CONTENTS
The set of integers
1. Set of integers «Z»
2. Ordering and comparing integers.
3. Adding and subtracting integers.
4. Multiplying and dividing integers.
5. Repeated multiplication.
6. Numerical patterns.
Equations and inequalities
1. Equation and inequality of the first degree.
2. Solving first degree equations in one unknown.
3. Solving first degree inequality in one unknown.
Geometry and measurement
1. Distance between two points in the coordinates plane.
2. Geometric transformations (Translation).
3. Area of the circle.
4. Lateral area and total area for each of the cube and the cuboid.
Statistics and probability
1. Representing the statistical data by using the circular sectors.
2. Random experiment.
3. Probability.
O TIMSS Questions.
O Glossary.
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى EE
The set of integers
ші
2
е,
=
z
2
دن
Set of integers "Z"
Ordering and comparing integers.
Adding and subtracting integers.
Multiplying and dividing integers.
Repeated multiplication.
Numerical patterns.
Н هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع خر
OE A T "-gG E:
: SP aun
ر0
4
p Set of integers " Ww
тә Á
You know that 0 is the smallest natural number. [3 9
Now; the question is : are there any numbers less than 0?
4 To find the answer; let us see the following examples :
—ЕЕЕЕЕЕ د م
* Temperature :
{ In Canada, sometimes the
Ww temperature records 30* C below
zero. In this case; you can say
that the temperature is — 30* C
v
2
A * Diving :
1 In Ras Mohammed (about 12 km. ў
V from Sharm El-Sheikh), the normal V
ё ( diving depth is 10 m. below sea level. a
e In this case; you can say that the 2
depth is — 10 m. ^)
ES 0
Е в) »
Eee يع My ral mi هذا العمل خاص بموقع
M LII كدي | (emm | الصف السادس الابتداتى |
The numbers - ЗО and - (О are not contained in the set of natural
numbers, these numbers are called negative numbers. Each of them is
less than zero.
The natural numbers and the negative numbers form together one set called
"the set of integers" and it is denoted by "Z"
ie.[ ®={-›-з3›-2,-1.,0,1,2›,3›-})
The set of negative — | f1 The set containing The set of positive
integers 27 where : ; the number zero integers Z* where :
-1,-2,-3,-4,- ge Ш Z*-(52,.3.4, ) |
| 2.8, Pol. ee
„рб Дз ET
As shown in the opposite figure.
The following diagram shows the relation between Z ,7* , Z^ and N
Í Set of negative Set of positive -
integers 2 integers Z *
Set of natural numbers IT
тат ابتدائي / تيرم ؟ V رياضيات لغات wall (э)
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى EE
r From the previous diagram , we deduce that :
]1[7 5 . Z'CZ , Z-CZ . (0) —Z
[222-NUZ-
ANZ =©
[4]Z-N=Z , N-Z=Ø
Remarks
The integer zero is neither positive nor negative.
i.e. 0 6 Z* andO z-
E The set of non-negative integers = {0,1,2 ,} = {0} UZ’ =
E] The set of non-positive integers = {0 ,- 125-39} = {0} U Z7
[El The set of odd integers = {~ ,-3.-1,1,3,}
E The set of even integers = 1 - .-4,-2,052,4,~}
Example [1)
Put the suitable sign "€ , É , C or ¢”:
112 (02 |x 13 (39)
610] ) 2- [901 lz
le] 2 С) РА [f] {022,5} С) 2
[01{2,-з}[ j» [һ]{6,-т}[ [5
2-а ши (а
[а] [b]e [c] Ide [el
19 Ig] c [hn] C шс Шс
m)
EE هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى
Try by yourself
Put the suitable sign “Є, E; C ог”:
[а]{0}[ )2* pl-7(. 2ل
[e]10{ )2- [4]{25,-4}[ [5
Write an integer to represent each of the following situations :
[a] A profit of L.E. 25 [b] A loss of L.E. 3
[c] 10 degrees below 0 [d] An increase of Р.Т. 75
[e] 6 m. above sea level. [f] 19 m. below ground.
[9] A building is 12 m. high. [h] 4 steps backward.
[a] 25 [b]-3
[c] - 10 [d] 75
[е]6 [f] – 19
[g] 12 [h]- 4
Try by yourself
Write an integer to represent each of the following situations :
[a] A temperature of 3 degrees below zero.
[b] A bank deposit of L.E. 100
[c] A loss of 5 yards in a football.
[d] A withdrawal of L.E. 25
[e] A decrease of 5 kg.
[f] A gain of 2000 pounds.
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى EE
Representation of the integers on the number line
( Every integer can be represented by one point on the number line ag follows : 3
Negative integers “ZT” The origin Positive integers “Z* "
- >
-5 -4 -3 Quim (NR 2 з 4
he negative integers } f The point that integers 1
are to the left of ¦ represente O ie called are to the right of
"the n". j
+ The set of integers is an infinite set, co it extende to infinity right to zero
and left to zero.
[Example (3)
Represent each of the following sets of numbers on the number line :
[a] {4 ,-2,0,3,-5}
[b] {-2,-1,0,1,2}
[c] {3 ,4,5,-}
Solution
TETE] هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى Й
Unit One
Opposites (inverses) and absolute value
On the number line , any two numbers that are at the same distance from O
and on two opposite positione of it are called opposites or inverses.
For example :
5 units
i Н |
-6 ЄБ)-4 - Ingo
Each of the integere 5 and — 5 has the гате distance away from О
Therefore 5 and — 5 are opposites.
i.e. The opposite of 5 ie - 5 and the opposite of - 5 is 5
The absolute value
The absolute value of a number is ite distance from О on the number line.
The absolute value of any | number x is denoted by ixl
The absolute value of any number (except 0) ie always positive.
` The absolute value of O ie O
For example:* | 4 |= + іс read ag : “The absolute value of 4 is 4”
“| - 4| = 4 ie read ag : “The absolute value of - 4 is 4”
١| O|=O0 ig read as : “The absolute value of O is O”
| Example (4
Write the opposite (inverse) of each of the following integers :
[a]-2 [b] 0 [c] 8 [d] - 33
Solution
[a] 2 [b] 0 [c]- 8 [d] 33
O N
22 هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى Й
Ө;
| Example (5)
Find each of the following :
[а]
[с]
[е]
[9]
Soluti
[a]]- 912 9
0-0
هيد|7-|+2- 8
-3|x|-2|=3x2=6
Ж om
Find each of the following :
[а] 1 +121=
[]141+1-51= ——
[е11-71х121= -——
| Example (6)
[b] 13]
[d] |-31+15]
[f11- 61-16]
[h]|- 101121
[b]I 31-7 3
[9]1-31+151=3+5=8
1#]1=61-161=6-6=0
[h]|- 101+121=10+2=5 —
[ь]18|=-
[d] | 10 | + |- 515 ———
[f] |- 151 41312 =
Find the value of x :
[а]1х1= 5 [b] |x|=10
Solution
[ce] |x|=0
[а] Since|x|=5 ,ћепх=5 or x=-5
[b] Since|x|=10 ,ћепх= 10 or x=-10
[c] Since | х|= 0 „then х= 0
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى EE
{ Exercise 1 Set of integers " Z"
( E From the school book
п Put ( v ) if the number is an integer :
ат O b Q
do C) e О
g 32( ) һ1 0)
В Put the suitable sign “€ , ¢ , Согд”:
a-3()N b {-5}()z
c Zero ( Z d (1,-2: н
е2 Oz f {3.4} Z Geasea2015)
08و )z h £31-651 J z^
i ANG) Z i {2,5,3} ()2
В complete :
(a) | - 5 |= *El-Kalyoubia 2013»
(b) If| x| = 5 then x = ——or-—— «El-Gharbia 2012»
(c) 3 + | 3 | > = «Cairo 2014»
(9) | - 17 |- 12 = + «El-Monofia 2017»
(e) ta (15) ...........-
«Suez 2015»
«Luxor 2012»
^3
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى EE
(El-Dakahlia 2017)
(El-Beheira 2014)
(Qena 2013)
(El-kalyoubia 2016)
(South Sinai 2013)
(El-Kalyoubia 2011)
(El-Menia 2012)
(Alexandria 2013)
Z-Z'UZz U (&-Sharkia 2014)
ZO ON = sss... (Port Said 2016)
EN The set of odd integers U the set of even integers = ==-
The complement of Z~ with respect to Z = --
The complement of Z * with respect to N = ---.......
В choose the correct answer:
(a) {2}: 2 "Giza 2014» (€ ог @ or
(b) 2 Z “Souhag 2015» (C or Є or
(e) | — 9 |... Z* «El-Beheira 2017» (E or É or Ф)
(a) 6-6 7, «E-Monofia2014» (€ ог @ or Ф)
(е)|-5|+3 2 “Alexandria 2016» ( or & or £)
MII: 7 (€ or # or or £)
(9)1-51+|7 |= = «South Sinai 2014» (12 or 2 ог -2 or -12)
(h)|-3|+|-2] = «El-Kalyoubia 2011» (— 5 or 5 or —1 or 1)
^w
TEE] هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى
Unit One
1-21+121= —— (North sinai 2017) (zero or 1 or —4 or 4)
If b = 1-7 | ,then b = ---------- (Е1-Мепіа 2011) (—7 or 7 or O or 14)
2027 =з (souhag 2012) (Z or Z* ог Z or Ø)
(El-Monofia 2011) (27 or Z* or N or {zero})
(matrouh 2017) (2* or {0} or Z- or 0)
(Et-Fayoum 2012) (Z or Z or N or Ø)
(The New valley 2021) (N or Z* or Ø or Z)
(81-мепіа 2016) (2 or N or Z or Z*)
(Kafr El-Sheikh 2012) (2 ог N or Z or Ø)
(assiut 2015) (27 or Z' or Z or N)
(Aswan 2013) (Z* or Ø or N or {0})
XC {2,-3}N {5-3} ,ћепх = <... (Giza 2011)
({2} or {-3} or {-5} or {5})
Write an integer to represent each situation :
a A temperature is 12 C° below zero.
She's diving 10 m. below sea level.
A temperature is 5 C° above zero.
EQ Ahmed withdraws 6000 pounds from his bank account.
The tree is 4 m. high.
3 steps forward.
80 ft. above sea level.
A bank deposit of L.E. 750
(rin Y تبرم / „дыд V رياضيات لغات уа] 62
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى EE
e
i Aloss of L.E. 20
1 Again of 7 kilograms.
k A profit of L.E. 100
١ A weight loss of 6 kg.
m A decrease of L.E. 200
a CA Complete the following using one of the words (positive - negative - zero) :
a Moving forwards is represented by numbers ; while moving
backwards is represented by numbers.
b Moving to the right is represented by numbers ; while moving to
the left is represented by ---.--.. numbers.
с Lowering than sea level is represented by numbers ; height
above sea level is represented by numbers.
d Sea level is represented by the number
E Represent each of the following on the number line :
a3,-4,1,-2 b -3,0,2,1,-6,5
с £26,-3,0,-1;3,5 d -4,—5,-6,-
e -2,—150.1.2 f -1,0,1,5-
B Write the opposite (inverse) of each integer :
а -3 b 12
d – 195 е |-34|
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى Tus]
Unit One
a Represent each number and its inverse on the number line :
b га-4
d M-99
Find:
а |-7| b |2] c 10]
а |-10| e |-21| f |- 100|
@ Find each of the following :
-3|*121 b [51-141
-2|*|-13I | d |—100|-|—50|
—5|*7 (Et-Menia 2016) f |- 121-1121
g 10|+]5| h [01+1-71
i |-2|+2 (Cairo 2016) j 1-51-5
Find each of the following :
а |-3|х|—5| b |—- 101x121 с |-30|+]—5|
d |-4|х|7| e |0|х|—3| f 8x|- 11]
Find the value of X :
a b |x|-2 12 c |x|=0
d e |3|=x | f |-101|]2 x
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى EE
@)
B £A Mark (true) or (false) and give the reason :
a Zer EZ”
Because :
Ø=Z NZ
Because :
Z'UN-Z*
Because :
[-17)€z
Because :
Z* is the set of counting numbers.
Because :
Zero is the smallest positive number.
Because :
In each of the following ; find the value of X to get a true statement :
a -4€ {7,x,-3}
b £3-5€ {-1,0,-3,х}
c хє{-2}
d tix€(2.5,-3) n (5.-2.-3]
e (3 42 : x1 U(- 4.0.4) 2 {0,-2,2,-4,4}
f |-5I£ (x.-5.3]
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع ( [сту
Ordering and comparing integers
9 The set of integers is represented on the number line as shown in the %
following figure :
1
Ascending order
v
<
From the number line above, notice that : 1
} n
2 * The numbers increase from left to right and decrease from right to left.
7
1 The numbers are in an ascending order from left to right. and they are in %
y a descending order from right to left.
i
5 ( For example : \
А: + | (which ig less than 4) is to the left of 4 А
* 3 (which is less than 7) is to the left of 7 9
5
ё
pue pom]
(2)
e This relationship holds true for all numbers on the number line, even
; when we go to the negative side.
For example :
+ —3is less than 2, because - З is to the left of 2
* - 5 іѕ Іеѕѕ than - З, because - 5 is to the left of - Э
| w-*—-9€—-2€-1«0e«1«2«3«-
Generally
For any two integers a and b, if the point representing a is to the left
of the point representing b; then a > b
Oooo
a b
For example : — 4 <-1 ee ee
-5 €4)-3 -2 (D0 1
because the point representing — 4 lies on the left of the point representing — 1
Similarly: «—5«2 2<3 2ه .0<1
EE
Q^» positive integer is greater than any negative integer.
Ө Zero is smaller than any positive integer and is greater than any
negative integer.
For example: O («)5 and O(>)-5
@ The least positive integer is “1” and we cannot determine the greatest
positive integer.
O The greatest negative integer is “— 1” and we cannot determine the least
negative integer.
)@
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى Tues]
Unit One
| Example (1
Put (<,>or=):
[al 4] [5 гы 5] ]-10 I] -12]__[-4
[a] ]ه )-2 te] -8] ]-7 if 1-31 js
tal |-4l{ )2 pll-si( Јо | 81-7] ]-I-e
Solution
[a] > | m> [c] >
[d] > | [e] > [f] =
[g] > [h] > [i]<
i arn
Put (< > or =) :
[a]6( J2 Ib]2( J-3 [e]0( )-1
Id]-4( )-8 [е] – 100 1 | If]l- 10] 9
Ig]l -51C3I- el mi sC Je | tij-I-41 C-2
Arrange the following integers once in an ascending order and
another in a descending order :
45,—5,1,-3,0,.6.-7and-1
Solution
Ascending order
"Descending order ——
The ascending order is: - 7 ,—5.—3.,—15,0.1.4and6
The descending order is:6 4و ,1.0.—1.,—3.,—5and-7
Сз)
EE هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى
(2)
[1] Arrange the following numbers in an ascending order :
45—3,6,0and-7
[2] Arrange the following numbers in a descending order :
-7 و 4 -و 0 و 9 -و 2 3800-1
Example (3 )
Write » using the listing method » each of the following sets :
[a] The set of integers greater than — 3
[b] The set of integers less than or equal to 2
[c] The set of integers more than — 7 and less than — 3
[d] The set of non-positive odd integers.
[e] X= {x:xEZ,x<-4}
[f]X={x:xEZ,-2<x<3}
Solution
[а] {-2 ,-1,0,1,2,--} [b] {2 ,1,0,-1,-2,--}
[с]{-6,-5,-4} [d] {-1,-3,-5,-}
[е]{-5,-6,-7,--} [f] {-2,-1,0,1, 2}
El-Moasser Application
"El-Moasser Maths 6th Prim. T2 "
Google Play © 5
ЖЕЕ] هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى K
2
Ordering and comparing integers
(EB Puti >or =]: d
a3) )-3 403
-3(J-4 -8()4
00-1 | -2()o
-12()-3 c33( )-6
i |-41C)2 ј |- 4IC)Iol
8( JI-8l 81-1315
-|ل)15- 2 | -6(_J-|-3]
ta-|-4l( 2 £33 «|-3l()8
а |-eI-I81C JI- 41 1-710)-7
Complete the following :
a The number is neither positive nor negative.
The smallest positive integer is and the greatest negative integer
is (Kafr El-Sheikh 2017)
The smallest non-negative integer is (Damietta 2016)
The largest non-positive integer is =-=-
The set of integers between — 3 and 2 = (Е1-Мепіа 2011)
The set of integers less than 1 and more than — 4 is { } Giza 2013)
aa (ОЗ)
Tus] ر هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى
موق کے
( Choose the correct answer :
a The smallest positive number is (Beni Suef 2015)
(zero or or 2 or 3)
=7 sain = |=9| (Damietta 2013)(> Or or < or <)
(74) |-4| (South Sinai 2013) (> or or = or 2)
An integer included between — 2 and 3 is (Giza 2017)
(3 or -3 or -4 or -1)
The greatest negative integer is ~- (El-Kalyoubia 2011)
(0 or -1 or – 100 or 1)
The integer which comes just before the number — 5 is
(Damietta 2011) (—6 or —4 or 4 or 6)
The integer which comes just next the number 23 is
(Et-Sharkia 2014) ) 25 or 22 or 23 or 24)
h The number of integers between — 2 and 2 = +. (El-Beheira 2012)
(2 or 3 or 4 or 5)
[а] Arrange іп an ascending order each of the following :
a 1,—5,-1and3 (El-Sharkia 2014)
—22,11,.-11.0and7
—75,-9,-4and-1
—35,5,2,-7,10and-6
8 ,- 62 ,- 19 ,- 42 and 0 (Beni Suef 2011)
-9,17 , ]- 9| :- 15 and 16 (Atexandria 2016)
-8 5:12 :|- 8 | :- 15 and 19 (Suez 2012)
£36,—60,2.—17,.-22andO
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى EE
Unit One
Arrange in a descending order each of the following :
a - 9 5.0 :7 3800-15 (red sea 2013) b 8 ,- 13 ,- 19 ,Оапа – 15
© - 28 ,- 35 , 33 ,- 37 and 2 d £21,-11,3,-1,-8and5
a Which of the following sets of integers is written from the smallest to the
greatest ?
а 0,14 ,- 15 , 1605-17 ,— 19 b 16.14.0.— 15 ,- 17 ,– 19
€ —19,— 17 ,— 15,0, 14 516 d None of the previous.
Which of the following sets of integers is written from the greatest to the
smallest ?
а 28,19,0.—-21.—36.—39 b —49,—38.0.22.27.30
с —52,—47 ,- 40 .0 ١16 .20 d None of the previous.
B £3 Write the previous integer and the next integer of each of the following
integers :
а -9 b 13 c 23 d zero
(E] C1 write the integers between each two integers of the following :
a-4,2 b -1,5 c —-7,0
(ED Write, using the listing method each of the following sets :
a Ш The set of integers greater than — 2
b The set of integers smaller than 0
C The set of integers greater than — 3 and smaller than 2
d C3 The set of integers between — 4 and 3 (Matrouh 2013)
e (The set of negative integers whose absolute value of each
is greater than 4
@
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى Tus]
(2)
f The set of non-negative integers.
g The set of non-positive even integers.
م ШХ={х:хЄ2,х<-2}
i X={x:xEZ,-1<x<1}
[11] Complete the following :
а —5,-4,-3, 600
t£2-75,-65,-5;-
4195,—45-0RÀ I» -—
BB =-2 05254 з,
-25,-—20,-15,
uA AREER BB
Write the letter that represents the integer on the number line :
A B C D E F ен ل K L M N OP а
-8 8
b -5
е 5
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى Tues]
Unit One
( Which point corresponds to the integer ?
E D G B
(14] Use the following table to answer the following questions :
City Highest temperature Lowest temperature
New York -2
Mosco
London
a Order the highest temperatures of all the cities from the greatest
to the smallest.
b Order the lowest temperatures of all the cities from the smallest
to the greatest.
Solve the following :
а Adiver is at — 5 т. and a balloon is at 5 m. Which is closer to sea level ?
b A helicopter is at an altitude of 1000 ft. › and a diving bell is at — 750 ft.
Which is further from sea level ?
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى EE
ee
MS
9 20 CA
ПШ, >
=
Adding and subtracting integers
ЭХ
First’ Adding integers 4
a Adding integers by using the number line $
The number line can help you in visualizing adding and subtracting integers. |
Just think of addition and eubtraction ae directione on the number line. |
> Ф 3-2
«2788 «C -
e To add a positive integer move to the right on the number line. — х
e То add a negative integer move to the left on the number line. «—— 8
с
а Adding two positive integers 1 |
v | n
Example a - = Y
20) 9 8 А
4 Use the number line to find the sum 3 + 5 f
Solution | IN
To find the sum 3 + 5 using the number line د do as follows : @
8 5 units V
= 0
5
* + + + + + poe + + + + ы
-5 -4 -3 -2 4 O | 2 (3) 4 5 6 7 )8( 9
• Start 3
e Move 5 units to the right.
This movement takes you to 8 So,3+5=8 —
Go)
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Unit One
The sum of two positive integers is a positive integer.
(Gl. Adding two negative integers |
| Example (2
Use the number line to find the sum (- 4) + (- 3)
Solution
To find the sum (- 4) + (- 3) using the number line ; do as follows :
3 units
E m
9 8 07) 6 544324101234
* Start at – 4
* Move 3 units to the left.
This movement takes you to (= 7) So, (— 4) + (-3) =-7
The sum of two negative integers is a negative integer.
ла
| Example (3 )
Use the number line to find the sum 7 * (- 8)
Solution
To find the sum 7 + (— 8) using the number line د do as follows :
Bunits а
а
4-3 220) 012 3 4 5 6(7) 8 9
e Start at 7
* Move 8 units to the left.
This movement takes you to (- 1) So.7*(-8)--1
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى EE
©
Example (4 )
Use the number line to find the sum (- 3) + 5
Solution 5 units
aL 2g cfe T ET
- 408-24 0
• Start at (- 3)
* Move 5 units to the right
This movement takes you to 2 So, (-3)+5=2
| Notice that:
The sign of the sum of two integers with different signs is the sign of
the integer with the largest absolute value. F
| | Adding integers without using the number line
? To add integers having the same sign, keep the same sign and add
the absolute value of each number.
For example :
+4ه. 7 11
(The sum of two positive integers is a positive integer)
*(-2)*(-6)--(-21*1-6)2-(2*6)2-8
(The sum of two negative integers is a negative integer)
To add integers with different signs, keep the sign of the number with
the largest absolute value and subtract the smallest absolute value from
the largest.
For example :
°7+(-1)=6
— The sum is a positive number because | 7 | > |- 1 |
— The sum is 6 because the difference of the absolute values of the two
integers is:|7|-1- 1127-1726
e(-6)+4=-2
— The sum is a negative number because |—6|>|4|
— The sum is — 2 because the difference of the absolute values of the two
integers is :|— 6 |—| 4| = 6-4 =2
)@
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى EE
Unit One
Possibility of addition in Z
From the previous examples, notice that the sum of two integers is
always an integer.
Generally
If we add any two elements of Z د the result will be an element of Z
It means that E Addition of two integers is always possible in Z
Example (5 )
Find the sum :
[а] (- 4) + C 5) [b]-2+6 [c]-7+1 [d] (- 9) + (- 6)
Solution
9-=)4+5(-= (5- | + 4-)-ء رو + )4( [а]
[b] -2 +6 +ع (1611-2) = + (6 - 2(2 + 4 - 4
[c] -7* 1--2(-7|-]10 2-(721) 6
[d] (— 9) + (-6)2-(-91*1-6) 2-(9*6)2- 15
Example ( 6 )
Find the sum :
[a] (- 2) + C 4) [b] -7*4 [Ic]-4*10 [d]8* (-3)
Solution |
[a] C2) + (-4)--6 [b] -7*4--3
[c] -4 * 10 2 6 [d] 8+(-3)=5
25р Calculator :
You can use a calculator to check your answers.
Example : – 78 + (- 105) = =~
КӘ 028) (+304 WU 83 |
бө: Y رياضيات لغات /\ ابتدائى / تيرم ука] Ө] (33)
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى [ЖЕЕ]
©
ЖШ arm
Find the sum :
[а] (— 5) + C 10) = —— [b]-8 + 2 = ..........
[с]7 + C 9) = [d] - 4 + 15 = ——
Properties of addition in
, а т Closure property ^
Zis a closed set under addition.
It means that the sum of any two elements of Z is always an element of Z
For example :
e-4€Z2 › -2Є7
;then:(-4)*(-2) 2 -6€z
Commutative property p @ تقر
If a and b are two integers , then : a + b = b +a
For example :
.-3+8=5 , 8+(-3)=5
ie. -3+8 =8 + )-3(
m @ Associative property m
If a , b and c are three integers , then : a + b + c = (a + b) + c = a + (b + с)
For example :
٠6+ (-4)+ (3) = [6 + (-4)] +(-3)=2+(-3)=-1
Also; 6 + (4) + )-3( = + [C4) + )-3([ = 6 + (- 7) -- 1
ie. 6+ )-4)+ رو = [6 + )-4([ + )-3( = 6 + ]-4( + C 9]
TETE] هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى D
Unit One
m 6 The existence of the additive identity (neutral) element іп Z <
For any integer a , we have :a+0=0+a=a
i.e. Zero is the additive identity element in Z
For example :
+2+0=0+2=2 +-3+0=0+(—3)=-3
( ш G The existence of additive inverse (opposite) property <
For every integer (a) there is an additive inverse (— a)
Where : a + (– а) = 0
For example :
* The additive inverse of З is — 3 , because 3 + (- 3) =0
* The additive inverse of — 4 is 4 , because — 4 + 4-0
* The additive inverse of zero is zero because O - O - O
• The additive inverse of a is (- a) and also the additive inverse of (- a) is a
i.e. The additive inverse of (- a) is - (- a) = a
For example :
The additive inverse of - б is - (- 6) = 6
| Example ( 7)
Use the properties of addition in Z to find :
[a] 8 * 10 * (- 8) [b] 24 + (- 19) + (- 24) + 9
Solution
[a] 8 + 10 + (- 8) = 8 + (– 8) + 0 (Commutative property)
= [8 + )- 8)] + 10 (Associative property)
=0+10 (Additive inverse property)
=10 (Additive identity)
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى EE
©
[b] 24 + (- 19) + (- 24) + 9 = 24 + (– 24) + (C 19) + 9 (Commutative property)
= [24 + (- 24)] + [(— 19) + 9] (Associative property)
=0 + (– 10) (Additive inverse property)
=-10 (Additive identity)
Example (8 )
دلا را {-3,6,3,-6,0}
[a] Find the relation between X and Z
[b] Is X closed under addition ? why ?
Solution
[a] X C Z because each element of the set X belongs to Z
[b] The idea of the solution :
We find the sum of each two elements in X , then if :
(1) all the results belong to X , then X is closed under addition.
(2) only one result does not belong to X ; then X is not closed under addition.
In this case :
Since ,6+3=9EX
So, X is not closed under addition.
Use the properties of addition in Z to find each of the following :
[a] 6 + 10 + (- 6) [b] 23 + (- 64) + 77 + (- 36)
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى Tus]
Unit One
Second Subtracting integers
{ Subtracting an integer means adding its opposite. 1
For example :
٠ Subtracting (— 2) from 8, means 8 c2)
adding the opposite of (— 2) to 8 |
[ The opposite of (- 2)i¢ 2] Keep Change Opposite
8 + (+2)
* Subtracting З from (= 5), means adding the opposite of 3 to (— 5)
(8) =(-5)+(-3)=(-8)
Keep Change Opposite 1
+ Subtracting (— 1) from (— 6). means adding the opposite of (— 1) to (- 6)
Possibility of subtraction in Z
From the previous examples, notice that the result of the eubtraction of two
integers ie always an integer.
eS
Generally
If we subtract any two elements of Z , the result will be an element
of Z
It means that : | Subtraction of two integers is always possible їп Z |
[ЖЕЕ] هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى Й
©
| p
Find the result of each of the following :
[a3]-5-2 [b]6 - 10 [c]0 - 6 [d]- 6 - (- 12)
Solution
[a]-5-2=-5+(-2)=-7 [b]6- 1026 + (- 10) 2- 4
[c]0-6=0+(-6)=-6 [d]- 6- (- 12) =-6 + (12) =6
| Example 10
Find the result of each of the following :
5-3 and 3-5
What do you notice ? what does that mean ?
Solution |
5-3=5+(-3)=2 , 3-5=3+(-5)=-2
We notice that: 5-3 #3 – 5
i.e. | Subtraction operation is not commutative in Z
Example (11)
Find the result of each of the following :
5-(8-1) and (5-3)-1
What do you notice ? what does that mean ?
Solution
5-—(3-1)=5-—[3+ )-1([ = 5-2 = 5 + )-2( =3
,)5-3(-1 = [5 + )-3([-1=2-1 =2 + )-1( =1
We notice that : 5 – (3 – 1) # (5 – 3) – 1
Le. | Subtraction operation is not associative in 2 )
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى EE
Unit One
From the previous , we can deduce the following properties of
subtraction in Z :
Z is closed under subtraction operation.
i.e. The result of subtracting any two integers is an integer.
The subtraction operation in Z is not commutative.
The subtraction operation in Z is not associative.
| Example (19)
Ifa=—-2,b= 3and c =- 1 , then find the value оѓ:
[a]a+b+c [b]a-c- (- b)
Solution
[аја+ь+с=-2 +3 + (– 1) = 1 + (– 1) = 0
[b] a-c-(—b)=-2-(-1)-(-3) =-2+1+3=-1+3=2
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى EE
a%-6 -5 -4 -3 -2 -1 0
с -3+(-4) d 6+(-7)
هد تيبب ببب ب مي С ЕЕЕ ЕЧ
|
-7 -6 -5 -4 -3 -2 -1 0 1 =2 -1 0 1 2 3 4 5 6 7
Use the number line to find :
a 5+2 b 4+(-3) с -4*(-2)
d -7+4 e 8-4 f 5-(-3)
9 6030-3-3 h -4+4 | i -10 +2
Find the result of each of the following :
а 4+2 b (-2)*(-1) | c -5+9
d 9+(-8) e 0+(-5) f 18+ (- 18)
g -6+0 h -48+34 | i —10 + (- 10)
a Find the result of each of the following :
a7-5 b 3-9 a-7-3
d 19—(— 11) (Giza 2013) e -3-(-4) f -9-8
820—7 h 0-(-3) i -5-0
j -73-(-73) k 33-|- 11] | 1 |-14|-]-28|
"هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى Tesi]
Unit One
Choose the correct answer :
a The additive identity in Z is (South Sinai 2011)
(0 or 1 or =1 or 2)
15 + 8—15 = eese (giza2015)(-15 or 8 or 15 or 23)
З=|=3| = (El-Gharbia 2017) (0 or 1 or 3 or 6)
1558 |= 5 (€l-Gharbia 2013)(1 or 6 or -6 or -2)
|-5|* 20 (€l-pakahtia2011)(—5 or 5 or 0 or 1)
f The additive inverse of (— 5) is (Beni Suef 2011)
(-10 or 5 or 0 or -5)
4*(-6)» (ismailia2011)(2 or O0 or —2 or —4)
If X =—1,Y > 2 , then the value of X + Y = ~- (Beni Suef 2016)
(2 or 3 or 1 or -1)
Write the property used in each of the following :
a -5+3=3+(-5)
b 6+(-6)=0
c 0+(-7)=-7
а (– 10 +5) +3 = – 10 + (5 + 3)
e -а+а= 0
Complete each of the following :
a4«(-3)-(-3)*
€ )-7( + =0 а (-8) + = (– 8)
е 6 + (– 6) = e (South Sinai 2011) f 2—(—3) = (Qena 2014)
9 |-17|-12= --------- (Е1-Мопоўа 2017) | h O + --- = |7 |
j (5 +(-8)) +7=5+(
(тыз / ايتدائى V رياضيات لغات wall
EE هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى
The additive inverse of 8 is
The additive inverse of (— 4) is - А (Ismailia 2014)
The additive inverse of | 6| is
The additive inverse of zero is (Souhag 2012)
The additive identity element in Z is (Souhag 2015)
The result of subtracting 7 from )- 2) is
The result of subtracting — 5 from 3 is ===-
Ifa +b = b +c ,then ce = ---- | S Ifa*(-3)-b*a,thenbz ع
t Ifa +b =b ,then a = u Ҥа+Ьь=0 »then a is
B Find the value of (n) in each of the following :
а -8+0=п | b -6+n=-6
€ n+6=0 d (-8)*(-4)7n
e 5+п=8 f -6+n=-9
9 27+(-27)=n h n+12=7
E] Use the properties of addition in Z to find :
a -5*(-6)*5 | b 10+(-5)+(-2)
с-7+2+(—13) d (— 17) + 19 + 17 (kafr El-Sheikh 2016)
е 15*(-3)*25 | f 5+(-3)+7+(-9)
25*(-8)*(-25)*7 (El-Monofia 2013)
h 55 + (— 255) + 45 + 225
— 74 + 65 + 74 + (- 65) (El-Beheira 2015)
113 — 120 * 17 (Red Sea 2011)
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى EE
Unit One
k £32015 + 180 + (— 1015)
| 63 + 54 + 37 +46 (Giza 2012)
@ Find each of the following :
a3+7+6 b (-6) + (-2) +(-1)
c -3+6 + (– 2) d 4-7-5
е -3+7-5 | f —17— 13 + 10
g -6-(-3)-5 | h -9+7-3
i -2+5-(-3)+1 j -10-4+(-3)-(-6)
к (-3+5)-(-6) ! -9-(4-7)
[11] Evaluate each expression for t=-12
а 45+t | b -12+t+24
c t* (- 21) | d t«|-8|«5
{Bf a=3,b=-4andc=-2, then find the value of :
a a+b b b+c
c a-b d b-c
e a+b+c f a-b+c
9 -c+a-b h (a*c)-b
£3 Check the property of closure of the addition and subtraction on the
following sets of numbers :
a Х= {1,0,1} b ү={-2,-1,0,1,2}
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى EE
©
Ga Temperature is recorded in St. Catherine — 3°C at three o'clock after
midnight , while it is recorded 11°C in the afternoon.
Calculate the increase in temperature.
@ The temperature on Sunday morning was — 2*C ; the temperature
dropped 7°С by Monday and then rose 5*C by Tuesday.
What was the temperature on Tuesday ?
[16] The temperature of the North polar water
layer is — 1°C, the temperature rises 5°C
in the North Atlantic deep water layer.
What is the temperature of that layer ?
CI A submarine at a depth of 90 metres
below sea level. It rose 60 metres.
Use the appropriate calculation to calculate
the new depth of the submarine.
B LI Ramy deposited a sum of money
amounting to L.E. 6220 , then he withdrew
an amount of L.E. 1211 , and then he
deposited an another amount of L.E. 2110 mass
How much is the balance of Ramy in the
bank ?
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى [сте
Multiplying and dividing integers
© First Multiplying integers %
n
{ You know that multiplication і a repeated addition. || 9 1
E. - Ec [
x For example : wae sane X
a 52072 IE به ¢9 X we 2 © vd
| 4 (0! 2 @ 4 5 © TA
9
در the sum of integers is always an integer; so the product of two integers ]
ie aleo an integer: ) n
2 For example : f
Ф ٠23 - 2 + 2 + 2 - 6 ع 2* a
f e )- 3( x 4 = )- 3) + )- 3) + )- 3) + )- 3( = - 12 EZ- و
of e 4x )-2( = )-2( + (-2) + (- 2) + (-2)=-8 EZ- V
e + ЄЗ)х(—2)=- (3 х(—2)) =- ((—2) + )- 2( + )- 2(( =- )- 6( = 6 ع 2* А
+0х3=0+0+0=0 9
4
©
(4)
(^ From the previous examples , notice that :
( 9 When we multiplied two integers; the product was always an integer.
Generally
If we multiply any two elements of 2; the result will be an element of Z
It means that : | Multiplication of two integers is always possible in Z
* When we multiplied two positive integers; the product was positive.
* When we multiplied two negative integers, the product was positive.
* When we multiply two integers one positive and the other negative;
the product was negative.
Generally
ө If the signs are the SAME then the product is POSITIVE.
ie. ©x@= © and ©) «© =(@%
Ф If the signs are DIFFERENT then the product is NEGATIVE.
{.е. @ х©=© and ©» © -©
e If we multiply any integer by 0 , the product will be O
For example :
°4x0=0 *0x3-0
| e-2x0=0 *0x-5-0
Example a )
Find the product for each of the following :
[a] = 8) x C 1) [b] 5x-2 [c] 0 x - 7
Id] 2 x1- 41 [e]-1-31x5 [f] -(-4) x6
Solution
[a] (-8) х (-1)=8 [b] 5 x-2=- 10
[с]0х-7=0 [d]2x|-4|/=2x4=8
[е]-1-31х5=-3х5 = – 15 [f]-(-4)x6=4x6=24
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى EE
Unit One
am ني
ER product for each of the following :
[a] -3 x 5 = <... [b] C 4) x C 6) = v~
[c] -9 x 02 —— [d] 6 x (-3) = :
[e] )- 100) x | - 2 | = .......... [f] )- 31) x 3 =
Properties of multiplication in
AA @ Closure property ^
Zis a closed set under multiplication.
It means that : The product of any two elements of Z is always an element of Z
For example :
2ت 3-ه 5-22
» then : - 3 х )- 2( -6 EZ
= @ Commutative property ^— —
If a and b are two integers , then : a x b =b xa
For example :
*(-3)x(-4)212 › (-4)x(-3)- 12
ie. -3)x )-4( = (C 4)x(-3)
^m © Associative property ^ =
If a »b and c are three integers , then: a x b xc = (a x b) x e = a x (b x c)
For example :
e4 x )- 3) x (~ 2) = (4 x )- 3(( x (= 2) = C 12) x (- 2) = 24
Also, 4 x (-3) x )-2( = 4 x )- 3( x )-2(( = 4 x 6 = 24
i.e. 4 x (= 3) x )- 2) = (4 x (—3)) x )- 2) = 4 x ((—3) x (— 2))
22 هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى Й
2
© @ The existence of the multiplicative identity (neutral) element in Z ^.
For any integer a , ме have:1xa=ax1=a
i.e. the number “1” is the multiplicative identity (neutral) element in Z
For example :
e1x3=3x1=3 e(—2)x1=1x(-2)=-2
-= © Multiplication is distributed over addition and subtraction in Z ^.
If a »b and c are three integers ; then:
eax(b+c)=axbtaxc апа (b+c)xa=bxatcxa
*ax(b-c)zaxb-axc and (b-c)xa=bxa-cxa
For example :
*2x(-4*7) *2x(-4)*2x7
22x89 | =-8+14
=6 =6
ie. 2x(-4+7)=2x(-4)+2x7=6
*3x(5-7) *e3x5-3x7
= 3 x )-2( | 215-21
=-6 =-6
ie. 3x(5-7)23x5-3x7 --6
Example ( 2)
Use the properties of multiplication of integers to find :
[a] — 4) x 57 x (- 25) [b] 8 x 2 x 125 x (- 50)
Solution
[a] (— 4) x 57 x (— 25) = (– 4) x )- 25) x 57 (Commutative property)
= (c 4) х (– 25)) x57 (Associative property)
= 100 x 57
= 5700
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى EE
Unit One
[b] 8 x 2 x 125 x (50) = 8 x 125 x 2 x (– 50) (Commutative property)
= (8 x 125) x (2 x(- 50)) (Associative property)
= 1000 х (— 100)
= — 100 000
| Example (3
Use the distribution property to find the value of each of the following :
[a] 3 (C4) *3x 5 [b]5x7 *5x (- 7)
[c] 15 x (C 17) + 35 x (- 17) – 50 x (- 17)
Solution
[a] 3 х (-4) + 3 x 5= 3 x ((- 4) + 5) =3 x1 53
[b] 5 x 7 + 5 x (- 7) - x (7 + (7)) =5 x0 =0
[c] 15 x (- 17) + 35 x (- 17) – 50 x (- 17)
= (15 + 35 — 50) x (- 17) = (50 — 50) x (- 17) = 0 x (- 17) 20
| Example (4
Find each of the following by two methods :
[a] 5 x (-3 + (—5)) [b] 120 x 19 + 120 x (- 19)
Solution
[a] — First method: ل Second method: —
5x(-3*(-5) 5х (-3+(-5))
= 5 x (-3)+5x(-5) =5 x (-8)
=- 15 + (– 25) = – 40 2-40
[b] — First method : — Second method: —
120 x 19 * 120 x (- 19) 120 x 19 + 120 x (— 19)
= 120 x (19 + (— 19)) = 2280 + (— 2280)
=120x0=0 =0
=
(Wer Y ابتدائى / تيرم V رياضيات لغات wall (49)
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى EE
2> Calculator :
You can use a calculator to check your answers.
Example : - 15 x (- 12) = ==
OUELJE
COED COED integers
Notice that :
°6+2=3 because 2x3-6 (3 6Z)
+-15+3=-—-5 because 3x(-5)=-15 (—5ЄЛ)
°18+(-9)=-2 because — (-9)x(-2)218 (-2€2Z)
e (-24) + (=4)=6 because (-4)x6--24 (6 6Z)
Since the result of the division 5 + Э ie not an integer » because there is no integer
multiplied by 3 gives 5 » so we can say that :
| The division is not always possible їп Z ог Z is not a closed set under division. |
Р
The following rules are applied when dividing integers is possible :
The quotient of two integers with the SAME sign is DOSITIVE.
و .ه.أ + © - © and ©+©=@®
For example :
+8+2=4 2(>5-)+10-ه.
The quotient of two integers with DIFFERENT signs is NEGATIVE.
and О+®= О ©= © ب قو .ه.أ
For example :
8-=)5-(+°40
TETE هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى Й
Unit One
Notice that :
0+2=0 60+ (1-4) =0 60 + (-100) = 0 7
ie.| the quotient of zero divided by any non zero integer is zero. |
Notice that :
Division by O has no meaning. Га
For example :
— 5 + 0 has no meaning because there is no number when multiplied by zero
gives (- 5)
Ж“ accom
State whether the quotient is positive › negative or 0 :
[a] -6 + (-2) Co) | [b]-10+5
[c120*(-5) . 0) [9]0 +3
[e] 0 + )- 5) ( ) [f] 144 + 12
| Example (5)
Find the result of each of the following :
6*(-3) and (-3)*6
Are the results equal ?
Solution
6 + )- 3) = - 2 while (— 3) + 6 is not possible in Z
So.6*(-3)2(-3)*6
It means that : [ The division operation in Z is not commutative. |
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى EE
(4)
| Example (6
Find the result of each of the following :
(36*(-6))*2 and 36+ ((-6)+2)
Are the results equal ?
Solution
(36 + )-6(( +2 36 + (76) + 2)
=(-6)+2 = 36 +(– 3)
=-3 =-12
So, (36 + (- 6)) + 2 * 36 + ((— 6) + 2)
It means that : | The division operation in Z is not associative.
From the previous , we can deduce the following properties of
division in Z :
Division is not always possible in Z or Z is not closed under division.
Division in Z is not commutative.
Division in Z is not associative.
Example (7 )
lfa-6.b-—2andc--6.then find the result of each of the following :
[a]4a*2b [b] (a x b) + c
[с] (a + c) +6 Id] (a - €) + b
Solution |
[ајда + 2 b = [4x6] + [2x - 2] = 24 + (- 4) - - 6
[b] (a x b) + c = [6 x )- 2)] + C6) =- 12 + (-6) 22
[c] (a + c) + b = [6 + (-6)] + (- 2) = 0 + (- 2) = 0
[d] (a - e) + b = [6- C6] + (=2)=(6+6)+(=2)=12+(-2)=-6 لس
(62)
5)
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى EE
<
/
(
exercise IJ
Multiplying and dividing integers
a 8x5
c -6x7
a (-15)*(-5)
c 0+8
LL From the school book
[1] State whether the product is positive , negative or 0 :
b (-3)x(-9)
d 0x(-5)
State whether the quotient is positive , negative or 0 :
b 24+(—3)
d 144 16
El Multiply :
a 3x5
€ (— 125) x )- 4) «&-Menia 2015)
e 9x(-1)
9 £3 (- 131) x (- 3)
i £1-(-6)x (C2)
k -|10|x|-3|
(GB Divide :
a 8+2
с 49 + (– 7)
е 0+ 10
9 – 100 +25
i (— 18) + (- 3) (&t-wonofia 2012)
k |-45|*1- 5]
-6x2
0x (= 10)
EQ -9x7
200 x (- 12)
(-5) x|-4]
101х141
(Ismailia 2016)
6
- 64 +8
(- 36) + (- 4)
77+ (- 11)
-18
2
18
(Aswan 2014)
-|42|]*6
ر هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى Teese]
(4)
Write the property of multiplication in the set Z in each of the following :
а – 12х1=- 12
b -5x (9х7) = (-5 х9) х7
с 5х (-2) = (2) х5
а (2х6) + (-2х 9) = - 2 х (6 + 9)
a Find the value of (X) in each of the following :
a-8x4=xx-8 -16xx=-16
€ xx(9+5)=(-4x9)+(—4x 5) -7xx=0
e M xx (5 x )- 13) = (- 9 x 5) x C 13) (8) x (-3)=x
-9+3= х Ф 8хх=-– 48
-3х= 27 @15х= 45
هل хх9=-– 45 -18+x=-9
Complete :
a The additive neutral element in Z is ·--------- › while the multiplicative
neutral element іп Z is ===
The sum of two negative integers is а ·--------· integer , while the product
of two negative integers is a =- integer.
The quotient of two integers having different signs when the division
operation is possible in Z is a -------- integer.
Bx =0
3 x (— 7) = ----- (Matrouh 2013)
|-24|+ (= 8) = eee (Ismaitia 2011)
5x(- |42 ) БЕ (Port Said 2015)
- )-12( x (-5)=
-4x [3 + (- 1)] . (Assiut 2016)
[9 + )- 5] x1- 111 = (The New Valley 2017)
-7x = 56
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى EE
Unit One
х9=-3х 21
If a = 3 ,b > - 2 ,then the value of 3 a = (Kafr El-Sheikh 2016)
Ifxz|-12],y2-3,thenx*ys ل (Giza 2017)
Ax (B + С) = e +АхС
If a طعا > a : 300 a #0 ,then b = ----------
Ifa +b =a , and a +0 ,then b = ----------
If a + b = 1 , then b&s-
Ifa + b = - 1 ,then b is the ofa
B Choose the correct answer :
а 6x(-3)2 ee (El-Menia 2012) (18 or —18 or 9 ог -9)
b )-8(+ )-4( - (El-Menia 2011) (2 or —2 or 4 or 32)
c 72+(-6)= (Et-Sharkia 2011) ( 12 or 12 or 6 or -6)
d
ҥх=|-2|›,у 3,then Xy = eese (Giza 2016)
(-5 or 5 or 6 or -6)
[8 + )- 3([ x C 3) = ----- (El-Monofia 2013)
(15 or —15 or 72 or -72)
f Zero + )- 3) = eee (4 or -3 or 1 or zero)
Zero х (— 1) x (— 2) x (—3) = === «Et-Gharbia 2014)
(0 or -6 or -5 or 6)
6-3x2-12-— (1 or 2 or 3 or 4)
If n is a negative integer; which of the following is the smallest ?
(Red Sea 2016) (З + п or Зп or n oF 3-n)
If a + b = zero where a +b , then a x b (Beni Suef 2017)
(= or > or > or 2)
( a Use the properties of multiplication of integers to find :
a 5x17x2 | b 50x(-45)x2
€ 4x(-5)x3x(-2) а (-2) х (-3) х5 х (1)
е 4х (- 16) x 25 f 8x77 х (- 125)
(s)
EE هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى
(0 Use the distributive property to find the result of each of the following :
a 3x(-2)*3x5
75 x 37 + 75 x 63 (El-Dakahlia 2017)
(-5)x(-6)*2x(-6)
147 x 69 — 47 x 69
112 x 17 * 112 x (- 17) (Souhag 2012)
(— 35) х (- 42) + (- 35) x 52
32x18—32x34 + 32x 17
45 x (— 16) + (- 47) x (- 16) + (— 16)
(-3)x4-(-3)x5-3
(KI) Use the distributive property to find :
a 26 x 101 | b 64x99
c 32x98 d 72x111
Find the result of each of the following :
а (—5)х(3+7) b 12 x (5-9)
с [8 + )- 5([ x 6 (et-katyoubia 2011) | d [5 + (C 3)] x (C 11) (Damietta 2013)
e 6x(- 6+0) (Damietta 2012) | f (-7)x(6-2-8)
g (5+3-8)x(-4) h (—5+3)+2
i 45+ ]3-)-6([ j p5xc2]*c59
(Bir x= 2 ,y=1 and z=5 , then find the value of : 3 x -2y +z
(Kafr El-Sheikh 2011)
( (EE Find the value of :X—2y +4 ,when X= 8 andy =-2 (cairo 2011)
(Birx=3 +y 7-1andz--2 , calculate the value of : (2 X + y) x3z
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى Tues]
(El-Dakahlia 2011)
Repeated multiplication
| You know hou to factorize a number by writing it ae the repeated multiplication. || м
) l y 9 م р |
For example : y
l6-2x2x2x2 Y
(2 multiplied by itself 4 times) „4
ie. 2 x 2 x 2 x 2 is another form of writing the number 16 1 a
9
• A third form of writing the number 16 is 24 37
i.e. Instead of writing 2 x 2 x 2 x 2 „ме can write 24 !
2 It is read as : )
4 "2 to the power 4" OR "2 to the fourth power" 7
Q 2 is called "the base" and 4 is called "the power" | 9
»"the exponent" or "the index" 1
Generally 4
If a is an integer and n Є2* , then a x a x a x -- to n times = а" е3
where a is called the base and п is called the power » index or exponent.
0 4
& 3 <١
ALD) Y ابتدائی / تيرم V/ رياضيات لغات жа] Ө] (57
| Esel os خاص بموقع ذاكرولى التعليمى ولا يسع بتداوله على مواق jen M
sated!) كتحابي Па 0590) «йу ЧЁ ميم
For example :
„7 х7 = 7? ,itis read as "7 to the power 2" OR "7 to the second power".
e 6 x6 x6 = 63 ,itis read as "6 to the power 3" OR "6 to the third power".
e (=3) x (-3) x (- 3) x (- 3) x (- 3) = (- 3(5 » itis read
as "— 3 to the power 5" OR "— 3 to the fifth power".
Notice that:
* The second power of a number is called the square of this number.
For example : 72 is read as "the square of 7"
* The third power of a number is called the cube of this number.
For example : 63 is read as "the cube of 6"
Remarks 4
o Any number to the first power is that number itself.
For example: • 9! = 9 • (-3) =-3 exizx
© Any number except 0 to the zero power is 1
For example: • 50 = 1 e(-79 = 1 „+ а0 = 1 where a «0
Ө If the base is one and пЄ2, then 1" = 1
For example: • 15 = 1 6112=1
(a) ifn is even
—(a)" if n is odd
i.e. • A negative integer raised to the power of an even integer gives
a positive integer.
Q 115 مومه 2ج EZ* , then - a)" =
* A negative integer raised to the power of an odd integer gives
a negative integer.
For example : • ( 4)? = 4? • (- 49 = – (4$
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى آ22
Unit One
| Example (1)
Find the value of each of the following :
[a] 25 Ib] )- 54 Ic] (– 3 Id] – (6)?
Solution
[а] 25 =2х2х2х2х2= 32
[b] (C552 54=5х5х5х 5 = 625
[с] )- 35 =- (3) = -(3x 3x 3) 2-27
[d] - (6)? = - (6 x 6) - 36
> Calculator :
You can use a calculator to check your answers.
Example : )- T) =
لتكت عانقا تا WOW
Find the value of each of the following :
[а] (- 5)? x 2 [b] )- 25 + C 2
Ic] A 1! + c 1)' [d] 32 + 3
Solution |
[a] (— 5)? x 22 = 52 x 22
= (5 x 5) х (2 x 2) = 25 x 4 = 100
[b] (- 25 + (- 3? = - (25 + 3?
=-(2х2х 2) + (3х3) =-8+9=1
[с] (-1)" +(-1 10 = — (1)" +0
=-1+1=0
[d] 32 + 33 = (3 x 3) + (3 x 3 x 3)
=9+27=36
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى EE
5)
Example (3
Ifa =3.,b=-1 апа с =2 , find the value of each of the following :
[a] a? + c3 [b] а2-2аь [c] (a - b*
Solution
[a] a? + сЗ = 32 + 23 = (3 x 3) + (2х2 х 2) 29 + 8 - 17
[b] a? -2 ab 232-2x3x(-1)29-(-6) =9 +6 = 15
[с] (a-b* = ]3 - C 1)] = [3 + 1? = 422 4x 4 = 16
Rules of powers
You know that:32=3x3 апа 35=3x3x3x3x3
S0, 22 x35- ($ х 3) x(à x x 3 x 9x 3)
2 356 351037 ات E
ie. 32 x3? =37*5 = 27
By using a similar way » you can find also that : (- 5)6 x ( 5(3 = (= 5)?
Generally
Ifa€z- (0) مء €Z* ,m EZ” , then:
i.e. In case of multiplying numbers with equal bases ; keep the base
and add the powers.
For example :
e 32 x 33 = 32* 3 = 35 = 243
e (C 2f x (- 2(2 = C 24 * 2 = (- 26» 26 = 4
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى EE
Unit One
«х5хх?=х5*7=х1?
+уху% хуз =у1*4*3 = ув
e à? x b? x a x b = (a? x а5) х (b? x b) = a8 x b?
e )- 2(3 x 2? =— (2)? x 22 =— 25 = — 32
Tit cmm
olf g
[a] 322. [b] — 7)? Еа
[с] 23 х 28 =.......... [d] (— 3? x ( 33 = <...
[e] 22 x (- 2? = че [f1 32 x - 3? = —
(58
csp CF
By using a similar way د you can find that :
Generally
If a is an integer anda #0 ,n CZ* , m €Z* , mz n. then:
i.e. To divide two numbers with equal bases; keep the base and subtract
the powers.
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى EE
5)
For example :
+@=25-9=2%=8
„з?
Cae = 8-6و - (2-9
=a 6=a3 where a #0
Example (4)
Find the value of each of the following :
27 x 24 (5)? х (– 59
[а] 5^ [b] C5y
зу
ET
3 ير 5
[с] 57; хт.
Solution
4 7*4 11
[а] 252 = = 2 = 20-8 = 23-8
Ace ETE TOTS _ كرت رمع
" 25 = )5( )5 ع [b] cere
US a= se 20 №
Га ах твае а 1“ [Remember that :
1914 D x -(ay-s--(ap--9 | a°=1,whereaz0
iF arn
Complete :
[b] С oie = pe T
34x38 _ оилор _
[c] $5 voice مسب | fatti Ey
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى EE
ГА From the school book
п Complete :
a7x7x7=7"
Two to the seventh power = 2
Ten to the twelfth power = ^
Four cubed =
Five squared
Seven to the zero power = --
23+ 2 Е (Cairo 2013)
Bzero - E (El-Kalyoubia 2014)
Cc Dee + (Ту? O E (El-Gharbia 2015)
E 1)10 +(- 1) mo... (El-Fayoum 2012)
am
ans where m ,n€Z*,m»n (Suez 2016)
n 35+ 33 = .......... (Matrouh 2013)
b
c
d
e
f
9
һ
i
j
k
|
о (- 49+ m 4y E (El-Beheira 2017)
Find the value of each of the following :
a2 b 3? c 5?
d 104 е (-7* f 27
9 (-6? h - (9) i 50
k (-1)°° lj c4»!
; ( (ring the value of each of the following :
а 22х23 b (10)? x (- 10)* с (-59 x5?
d -(2/x22 e bf xp? х7 ke
ر هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى [ШЕ]
ea Find the value of each of the following :
а 27 +25 | b 34433
с 59.56 d 0137 «31
e (-6) + (– 6) (Assiut 2012) f (-4)5«(-4f
g 69 )-8(5 + 3 h аб+а
3 ,а+0
Find the value of each of the following :
a 23x 32 b (C5)? x 22
d E (4)3 х (- 1)5 | e t£3(-5?x(- 1)" |
j C(-2)*+(-3)3| k ca 17+ o 1 (= 1(5 + ) 50
g -(4)?x(-2)8 | Һ 023 +22
m 24+33 2 | n 34x 32 x 22 x 3
(E Find the value of each of the following :
а 25x6?
5
37
c 3x3 (Suez 2015)
(Giza 2013) |
a (3) х (– 3)
ان
(North Sinai 2017)
— 3(4 x ) 5
g Cx (Souhag 2013)
2 × (— 5
p 21:8
(Aswan 2012)
(C20 x 24
k 972 (Ismaitia 2012)
(64)
с (– 1* x (2) (Giza 2014)
f (71x23
i ) 2(3 + 2
| o (-2 x (23
(El-Beheira 2014)
(El-Menia 2016)
; C2Yx(-25
210| — (Kafr El-Sheikh 2017)
a 96 x(- 99
92 х (95
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى EE
Unit One
(— 5)" x ) 58 —39 )-4(5
EPI Бут n = a a (El-Dakahlia 2015)
22+ 23
24
25 + 1(5 (South Sinai 2013)
Bose ti D "
Simplify each of the following to its simplest form :
6 3 12
-X9 whereas0 a
b ———5 whereaz0
а5 а? ха?
x8 Xx Fe له
© XB, x3 Where X#0 d SPST
B Simplify each of the following to its simplest form :
where Х #0
54x 33 (— 2)5х37
а —зт Px Coe
G4 x (= 3)? Х5х
Erg а у where X لاو #0
а L3 (C27 .(- 3* , (- 4 ‚(— 1(15 and 2
b 23,32 , ) 2(3 , (100)? and (- 1(5
[Orange in a descending order:
а )- 2(3 , )- 27 , )- 2P and (- 5
b 102, (-1)5 , 1000 and (1000)ê (Damietta 2013)
c £2 102, )- 1)5 , 100? , (- 10)? and 1000000
WPu »»0orz]:
зт Y رياضيات لغات /\ ابتدائى / تيرم wall
EE هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى
f 62 )-2
K 00192 +. £ 34
RE eas (-1)" (Damietta 2012)
m (-2f + (- 354 5
Choose the correct answer :
(€ or € or C ог $)
(€ or € or C or С)
(€ or € or C or £)
The additive inverse of (— 8)? is (8 or -8 or 1 or -1)
The additive inverse of (— 1)? is (1 or|-1 or 3 or -3)
32 × 3 = 3 (сга 2014) (5 or З or 2 or 1)
234.222... (Et-Beheira 2013) (2 of 8 or 16 or 32)
37 + 37 = (ismaitia 2014) (0 |or 1 or З or 7)
265 22 + 2 = (E-Gharbia2014) (28 of 212 or 25 or 2)
25+ 25-3 (€l-Dakahtia 2017) (2 or 10 or zero or 1)
(3)0 + ) 3)0 = --------. (South Sinai 2012) (6 |or O or 1 or 2)
(= 1(3 1 = (cairo 2017) (C2 | or O or 1 or 2)
22+2=.---- (6 or 8 or 23 or 43)
33-322... (3 or 35 or 39 or 18)
(= 3)3 (2392. (Souhag 2013) ((- 3)? ог (+3)® or -18 or 18)
324324322. (Aswan 2015) ( 28 or 49 or 33 or 29)
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى EE
Unit One
(5452 (5? or 20 or 15 or 30)
(—1)1 + (— 1)'91 = ......... (El-Monofia 2016) (zero or —1 ог 1 or -2(
(7 or 1 or 8 or 7?)
(25 or (—2)5 or 235 or (- 2)35)
(212 or 22 or 2x2? or 8)
If 35 + 3° = 39 , then a = = (4 or 5 or -5 or 0)
Which of the following is the nearest to : 11? + 9? ?
(22+ 18 or 211 + 29 or 120+ 20 or 120 + 80)
If X = 1 ,у=-2 , then the negative number in the following is
(Ei-Gharbia2013) (X +y? or x2-y or х?+у or X2+ y?)
ҤЕ is an odd number د then the even number in the following is
(Et-Sharkia 2016) (F2 or F2+F or 2F+1 or ЕЗ)
If عم 2 then find the value of each of the following :
an? b Зп” c n**5
d n3-1 е 1 f 2п5+1
[14] If a=2 and b = - 3, find the value of each of the following :
a 3a?b b 2a+3b с a?+b?+ab (Alexandria 2013)
If a= 32, b = 23 Find the value of : (a - b)!? (El-Gharbia 2015)
[16] Use the distributive property to calculate the value of each of the following :
a (17)? + 17 x 83 | b 33x23- (23?
|
c (27)? + 27 x (- 17) d (23)? + 23 x 78 — 23
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى EE
24 WWW zakrooly:comp
N
© Numerical patterns
{ Numerical pattern is a sequence of numbers according to a particular rule. y bl
For example : |
* The set of natural numbers N represents -3 LS v
a numerical pattern where “each number
б E ч " 345 6
is more than its preceding by one’
Ww • The set of even numbers = {0,2,4,6 ,... } represents a numerical
pattern where "each number is more than its preceding by 2"
» Also , the set of odd numbers = (1,2,5,7,..] represents a numerical
ví pattern where "each number is more than its preceding by 2"
A Pascal’s triangle
۹ F 4| --=—Row 0
e Pascal's triangle is one of the most
interesting numerical patterns. T -—TRow1
e In this triangle › we notice that each TA ROW dex
row begins and ends with number (1) 3 1---Row3 V
e After the second row › each number is Br d 1 ERNE هم
the sum of the two numbers just to the 1 5 1010 5 1----RowS5
с left and right of it in the row above. Pascal's triangle $
ЫЈ For example : 1 + 3 = 4 and is represented by the red triangle. 4
ES б) 2
М.
Ton] على مواقع أخرى Eius يسح Y بموقع ذاكرولى التعليمى E e: |
M Iouem] [кишу юу)
Unit One
* There are a lot of numerical patterns that we can get from Pascal's triangle
as shown in the following :
== = 5
1. The first diagonal ie of course just "I" ع and the next diagonal has the counting |
numbers (1,2, З... eto.) , the third diagonal has the triangular numbers. |
= a
1а
The sum of each гош < Counting РЕК
4 Triangular numbers
2. The sum of each row is twice of ite preceding or the eum of the numbers in an
row ig equal to 2 raised to ће nt power or 2" , where “n” ig the index number |
of the row. |
For example :
The sum of each row
• Try by helping your teacher to discover another numerical patterns in
Pascal's triangle.
—
(өө)
[сте هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى
©
Describing of the pattern
Means discovering the rule of the pattern and expressing it in words.
| Example (1
Describe each of the following patterns , then complete in the same
pattern :
[31153555 7;
[b] 95 85 , 75.65 ;
[c] 3.6.12 . 4
[d] 1-254 ,-8 16.
Solution
+2 +2 +2 +2 +2
[а] 1... م ac. do
(Description of the pattern )
Each number ie more than ite preceding by 2
-10 -10 -10 -10 -10
pres 8s OS 5
(Description of the pattern )
Each number ie lege than ite preceding by IO
x2 x2 x2 x2 x2
(Description of the pattern |)
Each number ie twice of ite preceding.
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى EE
Unit One
x(-2) x(-2 x(-2 x(-2 x(-2 x(-2
wit. es tae Coe
| Description of the pattern )
Each number equale ite preceding multiplied by (- 2)
Complete in the same pattern :
[a] 154 97 و 10 pee gien
[b] 2.8. 32;
[с]6,2,-2,
[d] 128 .64..32 "...و ©
| Example ( 2)
Write the number of line segments in each shape , then write
the numerical pattern and describe її:
[= CEL LIIT
Solution
Number of line segments : 6 , 9 , 12
The numerical pattern: 6 و 18 , 15 , 12 , 9و
Description of the pattern : each number is more than its preceding by 3
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى EE
the school book
E Complete in the same pattern :
a 1,2,3,4,
(Assiut 2017)
-85-5
-2,-4,-6,-8 s (Beni Suef 2013)
2,4.8,16,
b
c
d
e —-15,-125,-9; : (El-Sharkia 2011)
f
g
h
3,9,27,- . (Souhag 2012)
EB complete in the same pattern :
a 1,-1,-3,-
3,—-6.12,.—-24,-— "d (Alexandria 2011)
-6,-4,-2, + (Kafr El-Sheikh 2016)
9 > 6 > 3 5 0 goce و
16 .12 .8 4 (Ismailia 2011)
-3,9,-27
Complete in the same pattern :
2,-65185—54 ومسو
-5,-10,-15,-20,
8,848 3:
—20,-18,-16, (Aswan 2011)
e 1,3,6,10,
@ ١
TEE] ر هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى
Unit One
f 8,4,2,
g 1,1,2,3,5,8 (Aswan 2013)
a CA Complete the following numerical patterns by writing three consecutive
numbers :
a6,14,22,30,38,
b i 3 i 5 4 5 $ анисин (El-Beheira 2016)
€ 2953955958 9139 узше усе сын (Et-Katyoubia 2013)
d 1,4,9,16,29]8 m ER, amen
Discover the rule of the numerical pattern , then complete in the same
pattern :
а 160,80,40,20,
b 1.357,15;
€ 1945599 و 14و ------
d (n 4 P 2 LN $ ODE (Kafr El-Sheikh 2015)
a ШЇ Discover the rule of the numerical pattern and write the missing
numbers in each case :
(Aswan 2015)
РИО)
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى EE
©
( £ Complete the following table :
The numerical pattern Description of the pattern
3,57,511,515,19,23,
Each number is more than its
preceding by 5
Each number is less than its
preceding by 4
3:91
Æ Describe the pattern in words , then complete in the 18 1 pattern :
2.7.12 ,17,-
а
b 357511515 ...و
6,9,12,15,
81,76,71,66,
15359527 gee
f 192 ,96 ,48 ,24 و
[9] £ Write the number of line segments below each shape , and then write
the numerical pattern and describe it :
Number of line segments :
The numerical pattern : =
Description of the pattern :
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى EE
Unit One
@ EQ Write the number of triangles below each shape , and then write the
numerical pattern and describe it :
Number of triangles :
The numerical pattern :
Description of the pattern
Using the number of line segments , write another pattern and describe it.
[11] £3 Deduce the pattern rule expressing the following design , then write
the numerical pattern :
Number of line segments : ..........
The numerical pattern
The pattern rule :
CA Write the number of dots below each figure of the following :
Number of dots :
The numerical pattern : ~-
The rule of the pattern :
B Look at the pattern of dots :
We asi
а How many dots will be there in the 5'^ shape ?
b Draw the 5" shape in the pattern.
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى EE
( @ Look at the pattern of dots :
oer
a Draw the 5" shape in the pattern.
b How many dots will be there in the 7'^ shape +
Write the number of chords below each shape , then f nd the number of
chords that will be in the 6'^ shape :
ER An Egyptian land company reclaims 6 feddans per|day to become
prepared and ready for agriculture.
How many days do the company require to reclaim about 50 feddans ?
Write the numerical pattern which expresses this and describe it.
Karim saves L.E. 52 every month.
How many months does he need to save about L.E. 160 ?
Write the numerical pattern which expresses this and describe it.
B EA Khaled decided to lose weight at the rate of 3 kg. monthly.
If he is 90 kg. heavy right now , then how many months does he need to
reach 69 kg. ?
Write expressing the numerical pattern and describe it.
EE هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى
Test on Unit One
Answer the following questions :
п Choose the correct answer from the given ones :
а NUZ He (Z or Z* or Z or 2)
b The smallest positive integer is (0 or 1 or -1 or 2)
(> or > or = or 2)
5 280 + (— 5( 2686 =... (zero or 5 or 2 or 10)
Ifa =2 ,b=-3,then 2 ab = =- (-12 or -18 or -2 or 12)
Complete :
a 19- (=) = wees
b The additive inverse of the integer (- 17) is ............
© 2*02-
d (— 64) +8 = e
e The additive identity element іп Z is ....
a Arrange the following numbers in an ascending order :
-9,17 ,|- 9 | , - 15 and (- 4?
b Express each of the following sets using the listing method :
(1) The set of integers less than — 2
(2)X={x:xEZ,-3<xs2}
[ств هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى А
( a a Use the properties of addition in Z to find :
(1) C 15) + 23 + 15
(2) 36 + (- 72) + 64 + (- 28)
| C3*x35
b Find the value of the following in the simplest form at
[5] a Complete in the same pattern : |
(07,3,-1, Щ
1 1 1 1
(2) 3'68^12724^
b Use the properties of multiplication in Z to find :
(05x(4)x2
(2) 45 х 117 – 45 x 17
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع ( 1585251
Technology of Unit One
ё 5
le ^p.
ciel Using Excel program to find the sum of two integers 9
м !
6
1
Procedures of activity : &
[1] Click "Start" button from the task bar.
©
[2] From the menu “All programs” select “Microsoft Office”, then select 4,
"Microsoft Excel". | 4
[3] Fill in any column (Say A for Mic
example) by a set of integers UE ét Mes iet fames Took Dus Window Hep م
Ф 215072
as shown. 77] z0 -B م пж жиз. |
-n o RENE
P Do the same thing with NIE 3 = = | Ө:
column B “Figure (1)"
a
3
5
4
0
-8
10
-12
b
7
3
4
2
7
6
0 8
ا à 7
0
Om.
[4] To calculate the sum of each
% g
Os
number in column A and its
corresponding in column B ,
do as follows :
©
* Click cell C2
4 A e Type = А2 + B2 “Figure (2)" 2
Le
4 Fig. (2) $
$ = 2
6 (79)
` |1888| العمل خاص بموقع ذاكرولى التعليمى ولا یسح بنداوله على مواق أخرى H
\ ماص es) E) [Ge الصف السادس | 7
©
٠ Click Enter and you will
obtain the sum of the two
numbers "Figure (3)"
C2 till the pointer changes to the
form (+) » then perform (Auto fill) a
by copying the formula from cell
C2 to C8 by dragging, then we
obtain the shown figure.
“Figure (4)"
B
b
7
3
4
2
1
6
0
* Notice and deduce the signs rule of addition. Fig.
| Activity ©) Using Excel program to find the ЫЛЫС? of two integers
Procedures of activity : |
[1] Click "Start" button from the
task bar.
[2] From the menu "All programs"
select "Microsoft Office" , then
select "Microsoft Excel"
[3] Fill in any column (Say A for
example) by a set of integers
as shown. |
Do the same thing with к=
column B "Figure (1)” Fig. (1)
|
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى [сте
Unit One
[4] To find the product of each
number in colurhn A by its
corresponding in column B »
do as follows :
* Click cell C2
* Type = A2 + B2 “Figure (2)”
* Click Enter and you will
obtain the product of the
number in cell A2 by the
number in cell B2 “Figure (3)"
* To repeat the multiplication
operation on the remaining
numbers ; click C2 till the pointer
changes to the form (+) , then
perform (Auto fill) by copying
the formula from cell C2 to C8
by dragging ; then we obtain the
RELI
shown figure. “Figure (4)"
* Notice and deduce the signs rule of multiplication.
wall ( 81 رياضيات لغات /3 ابتدائى / تيرم ؟ OVD
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى EE
[О Use Excel program to find the quotient of two integers and print the sheet.
G Use Excel program to verify that : а? x а" = a" * and print the sheet.
G Use Excel program to verify that : а" + а" = a" 7~" , m > n ,a > 0 and print
the sheet.
@ £ Watch the weather forecast which describes the state of the weather in
some cities , and register some cities of temperature less than zero and
other cities of temperature greater than zero in the follawing table :
nl |
| Temperature | | |
* How many cities are of temperature less than zero ?
* Consider yourself a resident of one of the cities where temperature is greater
than zero » and you will travel to the city of OTE than zero.
е two cities.
(a) Calculate the difference in temperature between ti
(b) Describe the preparations needed to travel to this city.
© Copy the diagram , and arrange
the integers — 1 through — 19 in
the circles so that the sum of the
numbers on each line is — 30
E
TEE] هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى ١
{= Lessons of the unit :
1. Equation and inequality of the first degree.
а; Solving first degree equations іп one unknown.
3.Solving first degree inequality іп one unknown.
© Test on unit two,
© Technology of unit two.
© Activity of unit two.
% 4
By the end of this unit, student should be able to
* recognize the equation and the inequality. * recognize the properties of each of the
+ express symbolically each of the equation and equations and the inequalities.
the inequality.
Oe
a
25
UNIT AIMS
+ find the solution set of each of the first degree
* determine the degree of each of the equation equation in one unknown and the first degree
and the inequality. inequality in one unknown using the properties
А of each опе of them, and represent it on the
+ find the solution set of each of the equation and number line
the inequality using the substitution set method.
® 20
هذا اسل خاس يموق ii дыз يسع بتر اله عل ماق БШ...
| | الصف السادس الايتداكى | ERR سح
WWW:zakrooly:com
• The figure below represents a pair of scales.
I o
The other pan ; X
x One of its pans
Ww contains a bag contains x 4
of orange and a weight of 1 9
a weight of 2 kg. 5 ka. ! , |
f 1
v j n
A * If we denote the weight of the bag by x kg.» then the whole weight in f |
~ the left pan will be (х + 2) kg. 9
Ww • The weights in the two pans are equal when x + 2 = 5 , or when х= 3
9 (because 3 is the unique number if added to 2 gives 5) ч 1
ё ( It means that the weight of the bag of orange is 3 kg. when the two pans 3
contain equal weights. f.
9
8 4
ez 84 &
5 العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى EEE
u 4
Unit Two
o х + 2 = 5 is called "an equation".
© The letter "x" in the equation is called "the unknown" or "the variable".
€ The number "3" is called "the solution" of the equation x + 2 = 5, and
{3} is called "the solution set" (S.S.) of the equation.
o The solution of the equation is the number which "satisfies" the equation.
i.e. which makes the two sides of the equation equal. E
Definition : An equation is a mathematical statement that has two expressions
Separated by an equal sign. One or both of the expressions contains one |
unknown (ог тоге). |
E ы тик وي
Examples of equations :
+х+4=9 +8-3х=5
+х+2у=-1 +х+1=3-2х
+х?+5=14 x3 +4 x =0
Example (1
For the equation 2x + 1 = 7 و check each element of the set (1 و 2 > 3}
to find whether it is a solution of this equation or not.
Solution
e When x = 1 , the left side 2 x + 1 = 2 + 1 = 3 £ the right side (#7)
Therefore › 1 is not a solution of the equation.
> When x = 2 , the left side 2 x 1 = 4 + 1 = 5 # the right side ( 7)
Therefore ; 2 is not a solution of the equation.
* When x = 3 , the left side 2 x + 1 = 6 + 1 =7 = the right side
Therefore » 3 is a solution of the equation.
In the previous example ; the set (1 ,2 , 3} is called "substitution set". ]
@
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى EE
QD
Example (9
Find the solution set of the equation
xX + 2 = 5 if the substitution set is :
[а] (- 253.4] [b] (- 151.2)
Solution
Substitute in the left hand side of the equation for X by the elements of
the substitution set as follows :
Nofice that :
(>) means since
(.-.) means then or therefore р
[а] If the substitution setis [22 ,3 ,4}
e When X--2 ~. The left hand side = — 2 + 2 = zero #5
i.e. — 2 is not a solution to the equation.
e When X =3 -. The left hand side - 3 + 2 = 5
i.e. 3 is a solution to the equation.
» When X 24 — Notjeo that :
“The left hand side=4+2=625 | Thesolution setis a subset
i.e. 4 is not a solution to the equation. of the substitution set. р
г. The solution set of the equation is {3}
[b] If the substitution setis [- 1.1.2}
e When x= – 1 -. The left hand side =-1+2=1#5
i.e. — 1 is not a solution to the equation.
* When x = 1 -. The left hand side = 1 + 2 - 35
i.e. 1 is not a solution to the equation.
e When х= 2 -. The left hand side =2+2=4#5
i.e. 2 is not a solution to the equation.
A
~. All the elements of the substitution «v,
set do not satisfy the equation. (2bemember that :
i.e. the equation has no solution in Ø is the empty set (null set)
the substitution set [- 1 1 2) which has no elements.
- The S.S. = Ø
@
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى EE
Unit Two
If the substitution set is {2,8 ,— 1,5 »— 3} ; tick [v] in front of the
number which represents a solution to the equation : 2 x 32 7
»-1(7) 20 +5) „в ) .-30)
The inequality
| In the previous example ; if we add
2 kg. to the left pan in fig. (1) the
scales will be inclined as in the
opposite figure; then the left
side (X + 4) kg. is greater than
the right side (5 kg.)
Fig. (2)
T So; we can express that mathematically by
X+4>5
| ifweadd 3 kg. to the right pan in
fig. (1) » the scales will be inclined
as in the opposite figure , then the
left side (X + 2) kg. is less than the
right side (8 kg.)
ы So, we can express that mathematically by
X+2<8
Notice that :
The signs of the inequalitiy are :
e “<” read ав و" less than” * “>” read ас "is greater than"
e “<” read as "ie lese than or equal to" e © > " read as “is greater than or equal id
ps
(87)
"هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتد اوله على مواقع أخرى EE
©
* Each of these mathematical statements X + 4 > 5 апа X +2 < 8іѕ
called an inequality because there is a sign of inequality (“<" less than or
">" greater than) between the two sides.
Definition : An inequality is a mathematical statement that hae two expressions ^
separated by an inequality sign (« or >). One (or both) of the expressions
contains one unknown (or more).
Examples of inequalities :
ex+4>9 *8-3x«5
°х+2у>-1 +х+1>3-2х
Ехатр!е (3
Find the solution set of the inequality x + 1 > 5 if the substitution
setis(3.4.5.6)
Solution
Substitute in the left hand side of the inequality for X by the elements
of the substitution set as follows :
«When X=3 2. The left hand side = 3 + 1 = 4 is not greater than 5
i.e. 3 is not a solution to the inequality.
* When x =4 2. The left hand side = 4 + 1 = 5 is not greater than 5
i.e. 4 is not a solution to the inequality.
*When х=5 2. The left hand side = 5 + 1 = 6 is greater than 5
i.e. 5 is a solution to the inequality.
«When х=6 2. The left hand side = 6 + 1 = 7 is greater than 5
i.e. 6 is a solution to the inequality.
^ The 5.5. = [5.6]
(88
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى ЖЕЕ]
If the substitution setis {2 ,3 4 و 5 . 8] tick [v | in front of the
number which represents a solution to the inequality : 2 X — 5 » 1
*4() »3| 20 .5 *8(]
mem casas eae سسسب اوماد ةج TES кезет?
The degree of an equation (or an inequality) : It is determined by the highest |
power of ће unknown (symbol) in the equation or inequality |
ос ا
For example :
* 5 X + 2 = 7 is an equation of the first degree in one unknown X
» x2 + -عر 3 = 0 is an equation of the second degree in one unknown X
e 2Х +3 у= 515 an equation of the first degree in two unknowns X and y
e 4 22-7 > 515 an inequality of the first degree in one unknown X
Example (4
Determine which of the following is equation or inequality "Give reasons" :
[а]х-3=5 [ط] 2 2+6 [c]4y -7 > 1
Solution
[а] 2-3-5 is an equation because it contains a variable “x”
and contains the equality relation “=”
[b]2x+6 is neither equation nor inequality because it doesn't
contain the equality or the inequality relation.
[с] 4у- 7 <1 is an inequality because it contains a variable “у”
and contains an inequality relation “<”
(Y + (م Y یرم Д 5/ رياضيات لغات alse
EE هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى
Example (5
Mention the degree and the unknown / unknowns of each of the following :
[a] x * 1322
[c] 4? -4 x?-2
[e] 4 23-5 24-8
Solution
[a] X * 1322
[D] 3x?-7 =4
[c]4y? -4 x? 22
[Id] y-4<2
[e] 4 x?-5 x*-8
[f]7 x-3y25
[b] 3x?-7 =4 5
[d]y-4«2
[f]7x-3y25
an equation of 1% degree in one unknown (X)
an equation of 2" degree in one unknown (X)
an equation of 3 degree in two unknowns (X and y)
an inequality of 15t degree in one unknown (y)
an equation of 4'^ degree in one unknown (X)
an inequality of 1% degree in two unknowns (X and y)
Example (6
Express symbolically each of the following :
[a] X is less than - 4
[b] X is less than or equal to 3 and greater than — 1
Solution
[a] X«-4
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى EE
3 >1-[ط]
Equation and inequality of the first degree
LJ From the school book
п Determine which of the following represents an equation or an inequality
and give reasons :
a2x+1=5
d x=7+2
g 112х=24
b 2 3 2+ 2 -1
© &x>7-5
h 28+ 5
Determine the degree of each of the following :
a 1х-7=1
d (33x?-6-214
g ї1х-2у=5
J tax*-4x?-o0
b 4b-3=5
e 3х?+х+4=0
h (£23x-2<-2
k 4x*3y?»2
2x29
c x<-25
5 x2 30
Express symbolically each of the following :
a Xis less than - 3
b 1 xis less than or equal to 2
с 11 Xis greater than or equal to 3
d xis greater than — 4 and less than 1
e Li Xis less than or equal to 7 and greater than 1
f xis greater than or equal to — 2 and less than or equal to 5
( [а] Find the solution set of each of the following equations :
a x+7=10 if the substitution set is (1,3 5]
b دع 2+5 > 12 if the substitution setis {3 ,5 ,7 ,8} (Suez 2015)
E
ر هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى Tess]
M2x+1=5 if the substitution setis [- 1 2,0,2}
x+4=0 if the substitution set is [152 53.4}
mi4x-3=9 if the substitution set is {2 ,3 ,4}
2x-52-1 if the substitution set is {0 51,2 ,3} (£t-Sharkia 2011)
-2+3х=7 if the substitution set is {0,1,2}
1+2х= 16 if the substitution set is (7 ,8 ,9}
3x*5-2-4 if the substitution set is {0 »— 1 »— 2 »— 3}
2x+3=9 if the substitution setis {2 ,3 ,4} (Damietta 2016)
Find the solution set of each of the following inequalities :
а Шх+3<5 if the substitution setis (4 53 ,2 ,1 0}
(Et-Gharbia 2012)
х-4> 1 if the substitution set is {7 ,6 ,5 ,4}
2x-3>1 if the substitution setis {— 1,0,1 ,2,3}
023х-1>-2 ifthe substitution setis {-2.71,0,1,2}
3x*4s-2 if the substitution setis [- 1.0.1.2 ,3}
Ш-х+1<4 if the substitution setis [3,22 ,0,2 ,3}
t22x*4522 if the substitution setis [3,22 ,— 1,0,1}
5х-1>4 if the substitution set is [2 ,3 4 :5 ,6}
(Given that the substitution set is M = [1.2.3.7]: (Suez 2012)
a Find the solution set of the equation : 2 X —5 = 1
b Find the solution set of the inequality: X + 3 > 8
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى EE
NS
Unit Two
£ Considering the set of substitution is M = [-1 ,- 2 ,0 ,2} find the
( solution set of each of the following :
a2x+1=5 b x-3«-1
a Find the solution set of each of the following equations :
3(x-2)=- if the substitution set is {7 ,8 » 9}
2x+1=x-3 if the substitution set is {2 ,4 »— 1 »— 4}
с 12(х-3) = х+1 ifthe substitution setis {4 5,6,7}
Ххб=х+5 if the substitution set is {1,2,3 ,4 ,5}
e E 2x43 if the substitution set is (4 , 6 ,—2 ,0}
[9] Complete each of the following :
a The equation is a mathematical statement include: relation
between two sides.
The equation x + 3 = 5 is of the - ` degree. (El-Fayoum 2016)
с The equation x? — 3 = 6 is of the ——- degree. (Giza 2017)
d The solution set of the equation 2 X 1 = — 1 is if the substitution
setis {0,1,2,3}
The solution set of the inequality 3 X + 1 < 0 is if the substitution
setis 10 ,-3 ,3}
Choose the correct answer from those given :
a (2 Which of the following represents an equation ?
(a) x- 17 (b)22-7215 (c)x»-11 (d2x*3-7
b The equation x? — 4 x? = 0 is an equation of degree. (Beni suef 2015)
(a) first (b) second (c) third (d) fourth
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى EE
QD
с Lì The equation x? + 3 = 4 is of degree. (Luxor 2015)
(a) first (b) second (c) third (d) fourth
The number which satisfies the inequality : X < — 1 is (Ismailia 2017)
(a) zero (b) 1 (c)2 (d)-2
All the following numbers satisfy the inequality : X > — 3 except
(Assiut 2017)
(a) zero (b)-4 (c) -1 (d)-2
The greatest integer that satisfies the inequality : x
(Atexandria 2013)
(а) з (b) 5 (с) 8 (d) 6
The number that satisfies the inequality : X 2 > 3 is ===-
(Assjut 2012 , Port Said 2017)
(а) 3 (b) 4 (c) 5 (d) 6
The number — 5 is a solution to the equation ------- where the
substitution set is Z
(а) х-3= 2 (b)2x-1=9
(c) - 2x * 32 13 (9) 2+ 3 - 2 +ع 12
If 3 is a solution to the equation : 2 X — 4 = a then
(а) 3 (b) 2 (c)-2
The set of substitution is {1 ,2 ,3 ,4} , then the set of solution of the
equation x + 6 = 10 is ---------- (Alexandria 2016)
(a) {1} (b) {2} (c) {3} (d) {4}
If the substitution set is {2 ,— 1,3 ,4} , then the solution set of the
equation : 2 2+ 3 = З іѕ
(а) {0} (b) (- 1} (c) {3} (a) Ø
)94(
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتد اوله على مواقع أخرى EE
Solving first degree
equations in one unknown
4 i [2emember that а
@ The equation in which the highest exponent
is 1 and has only 1 unknown is called
a first degree equation in one unknown و 35 ~... d
=з
.2х+[-5
°3-xs-9
@ Solving an equation means finding the value of the unknown that
satisfies the equation › and this value is called "solution" of the
equation.
G In some cases ; the equation can be solved easily by using the
substitution method.
v
2
@ It is impossible to solve an equation by using the substitution method »
if the substitution set is an infinite set.
Oe
How to solve a first degree equation in one unknown without using the
1
4 ( substitution method ?
` The goal is to transform the equation so that the unknown is alone on one side
of the equation and a constant term on the other. (X = c)
S To do that; you need to know the following properties of the equations :
ez
95
— 685 Gad dy edi ty cesta al m
Properties of the equations
-
© An equation remains valid if Q An equation remains valid if
the same number is added to the same number is subtracted
both sides. from both abe
i.e. The same number can i.e. The same number can be
be added to each side of subtracted from each side of
an equation without changing an equation without changing
the solution of the equation. the solution of the equation.
If x-125
„then Х-1+ | =5+ 1
ie.X-6
© An equation remains valid if | @ An equation remains valid if
both sides are multiplied by the both sides are divided by the
same number. same non-zero number.
i.e. The same number can i.e. Each side of an equation can
be multiplied by each side of be divided by|the same non-zero
an equation without changing number without changing the
the solution of the equation. solution of the equation.
Generally
If a » b and c are three numbers; then we have the fol
@ Ifa=b.thenatc=btc
@ Ifa=b,thena-c=b-—c
@ ға=Ь ,ћепахс=Ь хс
@ ға = b معطا a + c = b + c where c # 0
ر هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى Tus]
Unit Two
The following examples show how to use the previous equality properties
to solve an equation of the first degree in one unknown :
‘Example (1
Find the solution set of each of the following equations in Z :
[a] <+ 5 - 4 [b] 5x=30 [c] 2 2-5 = 13
Solution
[a] - x 524 Another method
(Subtracting 5 from each of the two sides) You can imagine that 5
AX*5-524-5
Ax*0z-1 ET
7 The S.S. = (- 1)
Check the solution :
moved from L.H.S. to R.H.S.
and became — 5
Х+ 5 =4—> х=4-5=-1
PNE
Put X. | in the equation X + 5 = 4
“CHS. =-1+5=4=RHS v
[b] : 5 x = 30
(Dividing each of the two sides by 5)
„5х = 30
Pore
.. The S.S. = {6}
Ax26
[c] - 2x-52 13
(Adding 5 to each of the two sides)
2х-5+5=13+5
= 2xX=18
(Dividing each of the two sides by 2)
. 2X 218
M. 2
2 The S.S. = {9}
Ax-29
Another method
You can imagine that 2
moved from L.H.S. to R.H.S.
and became divisor.
2 ون دعر —x=18=9
х=9
О ИЕТ
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى EE
Example Q
Find in Z the solution set of the equation : = -5-3
Solution
du E — 5 = 3 (Adding 5 to each of the two sides)
-5+5=3+5
=8 (Multiplying each of the two sides by 2) ~. x 2 - 8 2
52
626 ^. The S.S. = (16) ل-
Example (3
Find the solution set of the equation : 2 X + 7 = 3 in each of N and Z
Solution
у 2 X + 7 = 3 (Subtracting 7 from each of the two sides)
2 2+ 77-7 - 3 7
.. 2 X = - 4 (Dividing each of the two sides by 2)
хт
у. The S.S. in = Ø
2 The S.S. inZ = (- 2) -
If we add a number to its double » we obtain 21 Find this number.
Solution |
Let the number be X ; then its double = 2 X
$о,х+2 х= 21 .. 3 X = 21 (Dividing each of the two sides by 3)
„3х = a ге у Then; the number is 7
° 3
ө Find the solution set of each of the following equations :
[b] 2 x + 11 2 3. wherex EZ
[a] x-5 =2 ,where XEN
è Two natural numbers, one of them is twice the other and
their sum is 108 Find the two numbers.
(эв)
Tues] هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى
{2 From the school book
[3| Find the solution set of each of the following equations in N :
ax+3=7 b Qy+8=19
d x-9=-5 ex+11=2
9 28 х = 32 uxor2012)| h 3y=27
пы Z-
0112-5 | ө”
ст-7=4
f х+1=|—3 | (Assiut 2016)
i 4Х=|—8| (Red еа 2014)
zo
I 0 45=3
Find the solution set of each of the following equations іп Z :
a Х+9=3 (Giza2017) | b ш 2-12 -0
d -4+x=-8 | e п+17 = | 131
g у- (5) = 3
j 2-х=9
h -4 دع - 4
k 7-m=12
© х+8=0
f m-(-3)=1
i 9 -<غز 18 (Assiut 2015)
| X25
Find the solution set of each of the following equations :
a3x-2=7 » where X EZ
Ш4х+1= 17 where XEN
Ш6х+7=25 where XEN
8x+12=4 where X EZ
32-13 - 6 where X EZ
5xX+2=-8 where X EN
2x-5=-21 where X EZ
5 2+ 4- 14 where X €z
(Giza 2013)
(Qena 2014)
(Beni Suef 2012)
(North Sinai 2017)
(99 )
ر هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى Teese]
®©
i 2x+9=-23 where X €z (Kafr EL-Sheikh 2016)
J 2y+16=24 where у €
к 8+2m=18 where m EN
1١13-22-9 where X EZ (Et-Fayoum 2015)
т4+3х=3? > where X EZ
n 23х-2=- 19 , wherex EZ
o -8-4x=16 » where x EN
p Y+2=-4 » where y EZ
q $-4-7 » where XEN
3x-14=|-16|
td2m-*12-26
2x+5=-27 (El-Sharkia 2011)
0221-15 = 8
2-2-5
4n-3=-7
m3x=8
mplete :
If x*527,thenX- --------
If 2+ 2 = | - 5 | : رهطأ X = -= (Е!-Ғауоит 2017)
If5 х= 10 ,then х= e (cairo 2013)
If 3 X— 3 = 12 , then xX = ---------- | (Suez 2014)
If x + 9 = 11 , then 7 X = e
a
b
c
d
e
f
g
Coi
a
b
c
d
e
f
If2y =8 ,then y + 3 = ~ (El-Dakahlia 2016)
Unit Two
9 1#3у=6 then 5y > с
h If4 x= 24 then E د oe
i If2a*3-15,then } a= و
j If (x + 1) is the additive inverse of (- 2) , then X = <... (Et-Dakahlia 2012)
k The S.S. of the equation x — 5 = 24 in 27 is
| The S.S. of the equation x — 3 = (6)? in Z is
Choose the correct answer from those given :
a The solution set of the equation x + 5 = 2 іп Zis { (Alexandria 2011)
(a)7 (b) -7 (c) 3 (d) -3
ÛJ |] 2+ 3 - 5 عد و 2: then the solution set is (Damietta 2013)
(a) {=3} (b) {5} (c) (- 5) (d) Ø
If zero Є (5. X - 3} , then х= (Et-Monofia 2017)
(a) zero (b)-5 (c) 3 (d)-3
The solution set for the equation 2 X — 1 = — 5 іп Z is ===-
(Е(-Ғауоит 2011)
@{-3} — رم {3Û} (о) {3} (a) {=2}
If 3 X + 9 = zero , then the solution set of the equation in Z is ----
(El-Fayoum 2012)
(a) {9} (b) {- 9} (c) {3} (9) {- 3}
The S.S. of the equation 4 x = — 16 in Nis
(а) Ø (b) (- 4} (c) {0} (9) {4}
If ع 2 +ع ]- 4| , then x = ---------- (Port Said 2016)
(a) -2 (b)2 (c) - 6 (d) 6
1] | - 4 | x X = 64 ,then х= (El-Fayoum 2015)
(a) - 16 (b) 16 (c) 6 (d)8
101
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى EE
If2x=2,then3x-1=
(a)2 (b) 3 (c) 4
If 2 X= 0 ,then X = ---------
(a) 2 (b) 3 (c) 5
If -4 > then X = =
(а) 1 (b)9 (c) 20
If 2 a b = 10 ,then 3 ab = .....
(a) 5 (b) 6 (c) 15
If5x -8X*2X*4X-7114.then5 X+ 3 >
(a) 3 (b) 35 (c) 47
(9) 5
(d) zero
(Red Sea 2011)
(d)- 1
(d) 30
(d)8x
The solution set of the equation : X + 3 = 12 is equal to the solution set
of the equation =-=-
(a) x-3-- 12 (b) x + (-3)2 12
(e) x- (- 3) = 12 (d) x- (- 3) 5-12
Юго х=18 Find the value of x + 1
ҥхх(7-(—2))=(—8х 9)х(—1) Find the value of X
11 Number when added to its triple becomes 72 Find the number.
(The New Valley 2017)
(El-Dakahlia 2011)
[jJ Three consecutive natural numbers whose sum is 213
What are these numbers ?
Шш Find the solution set of the equation :
3х-1=2х+5 ,where XEN
For Excellent Pupils
Find the solution set of the equation :
2 (Xx + 3) = 4 , where XE Z
402)
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى Tues]
3
~ وه
BAS 5©
(7 wm
QS
Solving first degree
inequality in one unknown
Properties of the inequalities b]
Inequalities have many properties in common with equations. ў
The following properties will be used in solving first degree inequalities in one 1
unknown :
-—— — -< — — - - » 4
@ The same number can be Ө The same number can be |
added to each side of an | subtracted from each side of @ ;
inequality without changing an inequality without changing 7 |
the solution of the inequality. the solution of the inequality.
( \
For A|
or example
o "A. 7e Ф 250732
f^ x-1>3 if х+1<5 Ё
ћепх-1+ 023+ (I hen Х+1- (1) <5- I ^
5 A (By adding | to each side) (By subtracting | from each side) 8 \
ё ( i.e. X» 4 and the S.S. (in Z) | Le. X<4 and the S.S. (in Z) C
pe |) 8145:6278 | i5(352;1.05—-1.-25..] و
т 1 3 ^n
8 4
ez 103 :
ي هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى ges]
ian T ساس W asoca SE on
KO)
fa N
© The same positive number can @ Each side of ап inequality can
be multiplied by each side of be divided by same positive
an inequality without changing number without changing the
the solution of the inequality. solution of the|inequality.
f 8x«-5 22-2 م
„воп Xx 2-202 ‚бел 3 + B «-15 3
(By multiplying each side by 2) (By dividing each side by Э)
i.e. X>- 4 and the S.S. (іп Z) i.e. X<- 5 and the S.S. (іп Z)
is (-3,-2,-1,0 ЭШИ is [6 ,-7,-8,-9,-10,..}
© it we multiply or divide each side of an inequality by a negative number
we must revers the inequality.
-2Х > 6 (+-2)
fevers
lf -2X>6,then -2X+-2 <6+ -2)
2 e < 3
(By dividing each side by — 2)
i.e. X«- 8 and the S.S. (in Z) is ]- 4 ,- 5 ,-6 ,- 7,28...)
——
Generally
Assuming that a » b and c are three integers د then :
@ Ifa <b ,then a+c<b +c
@ Ifa <b ,ћепа-с<ь-с
@ If a <b ,c is a positive number ; then ac < bc
@ ға > b ,cis a positive number ; then i « 2
(© ға <b , cis a negative number ; then ac > bc
Ifa «b; cis a negative number , then = > 2 ©
Чоў
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى Tus]
| Example (1)
Find in Z the solution set of each of the following inequalities :
[a] x*7«9 [b]2x-525
then represent the solution set on the number line.
Solution
[a] x + 7 > 9 (Subtracting 7 from each of the two sides)
MAxt7-T7T«9-7 E2
.. The 5.5.= {1 -Ommel 2al =~ ee 1 2 5 4
"2x-52 5 (Adding 5 to each of the two sides)
A2x-5452545
-. 2x 2 10 (Dividing each of the two sides by 2)
. 2X, 10 з
a = a 0275
^ The S.S. = {5,6,7,8,--}
D-——' Q4 — M —H EN A—T7 raad
о 172 39 4 omen 009 10 110 12
Example (9 -
Find the solution set of the inequality : 2 x + 5 < 11 , where:
[a] xez [b] xEN
then represent the S.S. on the number line in each case.
Solution
‘w 2 х + 5 > 11 (Subtracting 5 from each of the two sides)
:. 2 2+ 5-5 < 11-5 г. 2X« 6 (Dividing each of the two sides by 2)
>£ “x<3
[a] When x EZ:
The solution set is all the integers which are less than 3
їе. Тһе 5.5. = {2,1,0,-1,-2,-} "^ —
Oe
[b] When x EWN: hod RM C ш:
The solution set is all the natural numbers which are less than 3
& ooon 2
i.e. The S.S. = {2,1,0} TOVM Wd Y 2 s» X 5
پچ
)09( الام ينحنت дейш ر is
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى EE
iG
| Example (3
Find the S.S. of the inequality : 4 2 x 2 10 . where :
[a] x En [b] xez
Solution |
у 4-2 X> 10 (Subtracting 4 from each of the two sides)
14-2 х-42 10-4
2-2 x> 6 (Dividing each of the two sides by – 2)
Ern Eod
500-25-2 The change of the direction
Axs-3 of the inequality sign.
[a] When x € N:
The S.S. is all the natural numbers which are less than or equal to — 3
i.e. The S.S. = Ø
[b] When xE Z:
The S.S. is all the integers which are less than or equal to – 3
i.e. The S.S. = (-3,-4,-5,-]
Find the solution set of each of the following inequalities :
[a] 2x-325,wherexE 2
[b] 1-2 x> 7 ,where xE 2
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى EE
The value of x can be between two integers , so the inequality can be
written on one of the following forms :
Q1«x«5
„that means: X> 1 апа x<5
Therefore : x = 2 or 3 4
© - 2 > 3
»thatmeans:x2-2 and xs3 8 2 4
Therefore : x = – 2 or = 1 or 0 or 1 or 2 or 3
@-3<xs1
„that means :x>-3 and xs1
Therefore : x = – 2 or — 1 or 0 or 1
@-1<х<2
»that means: x2—1 and х<2
Therefore : x = — 1 or 0 or 1
NS
‘example (4.
Find іп Z the solution set of the inequality : - 7 =2х- 5 < 1
Solution
71-752 X— 5 < 1 (Adding 5 to each of the three sides)
A-—7*582x-545«145
.. — 2 < 2 x <6 (Dividing each of the three sides by 2)
ANE
2 The S.S. = (- 1.0.1.2)
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى EE
9
Solving first degree inequality in one unknown
ШЇ From ће school book
[1] Find іп the solution set of each of the following inequalities , then
represent the solution set on the number line :
a LU x-3<1 (El-Fayoum 2015) | b х+2>5
x+4>1 d x+4<7 (Port Said 2016)
x+326 x-4<-1
f
19 <а + 14 h m-52|-7|
j
-1>х+3 3x<12
4kz-16 1 -2y«-14
Fina the solution set of the inequality : X + 3 > 6 , where :
а xez b xc 8
then represent the solution set on the number line.
Find the S.S. of each of the following inequalities د theh represent
the S.S. on the number line :
a LJ2x*1«7 ,мһеехє н
b L32x-3«5 , where XE 2 йй
c 3x-2«1 » where xE N
d 2x+9<1 »wherexe€ 2 (El-Sharkia 2016)
4x*22—10 , where xE 2
3x+9>0 > where xE N
3x-5s4 » where xE 2 (South Sinai 2017)
,wherexe 2 5-7 22-5 ل
ر هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى Tes]
[
Unit Two
i L44x*1«13 ,wherexc 2 (El-Kalyoubia 2017)
j 013х+2=11 .wherex€ 2 (Red Sea 2016)
k 9-6x<15 > where xE 2
11+2xs-3 > where xE N
m 0 3 2+ 2< 12 ,wherexEN
n 3x-5s7 » where x € Z* (Cairo 2017)
o >غ2 1-8 ]ا 33 , where XE Z (Suez 2015)
р 1-3х>7 > where X € N
[E Find the solution set of the inequality : 3 x + 5 > 2 and represent it on the
number line if :
a xEN b xez (South Sinai 2012)
Find the S.S. of each of the following inequalities . then represent
the S.S. on the number line :
a3<x+2<6 ,wherex €i b -3sx-1<3 ,where X EZ
€ 3> 22+ 159 ,wherexEN d 5غ 1-2 ك5 11 ,where x EZ
Complete :
If xX > y s then X + z
IfxX >y ,then X—z
If x > y and zis positive » then XZ : yZ
If x < y and zis negative » then х2 ------------ “yz
CO f x+ 5> 2 و then X> er (Alexandria 2015)
Ifa — 3 <0 , ћеп
i>
(109,
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى EE
1] 2 2-1 <7 معطاء 2 Xz
60 |] 3 معطاء 8 15 -ع : 3 <>
The S.S. of the inequality : 4 x « 8 in N is
The S.S. of the inequality : 2 x 3 > 5 in Z is
The S.S. of the inequality : 4 x 1 > 5 in Z is --
The S.S. of the inequality : 1 = x > 4 in Nis
m The S.S. of the inequality : - 2 > x < 0 in N is
n Ifb < 0 ,then b + З. З
E Choose the correct answer :
a The solution set of the inequality : x > 0 іп Z is (Ismailia 2012)
(а) 2 (6) Z* (с) Z- (d) N
The S.S. of the inequality : - 2 X < 0 in Z is ===
(а) 2 (b) (c) Z- (d)Z*
If عر € N , then the S.S. of the inequality : - x > 3 is ===
(а) {4:5:6 >=} (b) {-4,-5,-6,-}
(с) {- 3} (d) Ø
The S.S. of the inequality: 2 x 1 < 5 іп N is
(а) {2,1,0,-1,-2,-} (b) {2,1,0}
(с) {1 ,0,-1,-2, үх (d) {1,0}
ШФ If 2 +ع 5 > Запа xE 2, ћеп the solution set = =- (Damietta 2012)
(a) N (b) N- (0) (c) 2- (9) 2+
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى EE
Unit Two
The S.S. of the inequality : 4 — x » 3 in Z* is
(a) {0,-1,-2,-3,-} (b) {05152535}
(c) {0} (d) Ø
The S.S. of the inequality : – 1 < x < 1 in Z is (South Sinai 2013)
(а) {-1, 0} (b) {0 >t} (c) {0} (d) {1}
The solution set of the inequality : 2 < x > 3 where x EN is
(Damietta 2017)
(a) {zero} (b) {2} (c) {3} (d) {2 53}
If x»5,then:— Хз
(a) «-9 (b)2-5 (c) «-5 (d)>-5
The greatest integer that satisfies the inequality : 3 x « 6
(Aswan 2015)
(b) 4 (c) 5 (d) 6
2 belongs to the S.S. of the inequality: , where x E Z
(a x»2 (b)x«2 (c)-x>-3 (d)-x»3
For Excellent Pupils
B Find the solution set of the inequality :
2x-1sx-*3,wherexcz
هذا العمل خاص بموقع ذاكرولى التعليمى ولا یسمح بتداوله على مواقع أخرى FEE]
Test on Unit Two
Answer the following questions :
[1] Choose the correct answer :
a Which of the following represents an equation ?
(x+8 or 10-426 or x29 or x+1=5)
b The equation x? + x = — 2 is an equation of
(first or second or third or fourth )
с All the following numbers satisfy the inequality : x > — 2 except -
(—1 or 0 or 1 or -3)
d Ifx +4 =7 sthen 2 X = ----- (3 or 4 or 6 or -3)
е Ifa <b ,then —2 a -------- -2b (> or < or = or <)
Determine the degree of each of the following :
(1)x?+8=0 (2)2х-1>7
Express symbolically each of the following :
(1) x is less than or equal to - 4
(2) x is greater than — 3 and smaller than 7
Find the solution set of the equation :
3 x+ 2 = 8 if the substitution setis [- 1,0,1 ,2}
Find the solution set of each of the following equations :
(1)4x-3=5 ,.wherex€m
(22x*721 »wherexGZ
Unit Two
a a Find the solution set of the inequality :
3X + 2 < 2 if the substitution setis [C2 ,—1,0 , 13
b Find the solution set of each of the following inequalities , then
represent the solution set on the number line :
(1))5x-1»-11wherexcz
(2-1s2x*3«|-7| wherexEZ
a Find the solution set of each of the following in N and in Z :
(1)3x+1=-5
(24x-9«-1
b Anumber when added to its double becomes 15 Find the number.
Qe :0( Y ابتداتى / تيرم V رياضيات لغات yaks (113)
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى FEE]
| Activity ) Using Excel program to find the solution set of the first
degree equation in one unknown
Example &
9 20 CA
шї ©
0
Find the solution set of the equation: 3 X + 2 = 11 , if the substitution set
Ww is {-2,0,3,2} 4n;
<
N Procedures of activity : |
RS [1] Click "Start" button from the task bar. Wd
9 [2] From the menu "All Programs" select "Microsoft Office". then select 73
"Microsoft Excel". ۹
у B
[3] Write in any column (say B
for example) the symbol x
in the cell B1 then write the
elements of the substitution
set {- 2,0,3 ,2} below X
as in fig. (1)
Oe.
[4] In column C write
“3 x + 2" in the cell C1 ,
мэ g
Or.
9 then click cell C2 e
( [5] Туре 3* 82 + 2 5 x 0 1
2 4
t e [6] Click enter to see the result л Ч Co A
«6 —4asin fig. (2) 3 2 $
X Fig. (2) 4 i
= 114 :
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى |182 |
لست سس W asoca SE on
Unit Two
[7] To repeat this operation on
the remaining numbers ›
click cell C2 till the pointer
changes to the form (+) »
then perform (Auto fill) by
copying the formula from cell
C2 to C5 by dragging: then
you obtain the shown figure
"Fig. 3"
[8] From the data on the
Screen , it is clear that X = 3
satisfies the result 11
Le. the S.S. = {3} Fig. (4)
AEN
(115)
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى EE
ريك الال | عست 7
2
E
Activity of Unit Two )
5
e.
r3 a Use Excel program to find the solution set of the equation : 4 X — 3 = 17 ^
м
6 if the substitution setis {2 5 .6 .7] ү
@ Below each balance, express the suitable mathematical statement , &
\
then solve it in її:
Ww [a] | " ү,
) ё - The mathematical statement - The mathematical statement
- The solution set 5 . - The solution set
[4]
- The mathematical statement - The mathematical statement
© € — ——— аө, iR dec init rec Ui
- The solution set - - The solution set =-=-
\
e 2
$
4
2
a 116
ae till
M oe | (es | الصف السادس الابتداتى |
0
X
N
ê
|
1. Distance between two points in the coordinates plane.
2. Geometric transformations (Translation).
D 3. Area of the circle.
v
Ww Lessons of the unit:
4. Lateral area and total area for each of the cube and the cuboid.
© Test on unit three. © Technology of unit three.
© Activity of unit three.
ame
of the cube and the cuboid.
2 By the end of this unit, student should be able to 1
= ‘find the distance between two points on a ray • йпа the image of each of a point, a line
< and on a straight line. segment and a geometric shape by a given
j ‘graph the points in the coordinate plane. translation in the- coordinate: plane.
Ў 25 find the distance between two points in the * find the area of the circle
1 5 coordinate plane. * find the lateral area and the total area for each
ё + remember the geometric trasformations dhente andithe eubcid:
E (Reflection - Translation - Rotation) * solve word problems on the area of the circle
je s recognize the definifion of the translation and the lateral area and the total area for each
* find the image of each of a point, a line segment and
geometric shape by a given translation in the plane.
2g
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتد اوله على مواقع أخرى ]1825
D
Крит” (ы сез» on ساس in Н /
=
Lu Ca
оо
«Xo 2
U the distance between two points оп а ray 4
9 4 The distance between the two points А and B on a horizontal ray or Ө:
vertical гау = AB where : )
The length of АВ = number of the ending point number of the starting point \'
Am.
118
=В-А 7
For example к
D [1] In the following figure : [2] In the following figure : 0
|
vi PTGiA44 AG XT ib. [
19 If A represents the number З If M represents the number 2 K$? f А
; B represents the number 7 »N represents the number 7 А @
and C represents the number 10 and К represents the number 9 +6 9
^ 2 „then „then i gj 1
4\ AB = B-A=7-3= 4 units. MN-N-M-7-2-5units. |; 5
je ,BC- С-В 10- 7 = 3 units. \NK=K-N=9-7=2unts vi: | MB
56 and AC = С-А = 10-3 =7 units. |and MK-K-M-9-2- 7 units. | ۵
6 |
:
ر هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع =ч.
ENN cane aama
Unit Three
El The distance between two points on a straight line
* The straight line which mentioned here is the integers number line
whether horizontal or vertical. As you know it is an enlargement of the ray
of the natural numbers by adding Z
* The distance between the two points A and B on a horizontal line or
a vertical line = AB where :
The length of AB =| number of the ending point – number of the starting point |
=|в-А|
For example :
[1] In the following figure : [2] In the following figure :
AA в و
r3 43-10 ! 208 401896 7 8
If A represents the number — 3 If E represents the number — 4 r4
» B represents the number 4 » D represents the number — 1
and C represents the number 7 and F represents the number 5
»then > then
АВ = |4-(—3)| ED=|-1-(4)|
= |4 + 3|= 7 units. =|-1+4]=3 units.
»BC=|7-4| , DF =|5-(—1)|
= 3 units. = |5 + 1| = 6 units.
and AC = |7 - (- 3) and EF = [5 – (- 4)
= |7 + 3|» 10 units. =|5 + 4| = 9 units.
(119)
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى EE
!
Example a
In the opposite figure :
If the points A د B › C and D represent
the numbers — 7 ,—3 › 0 and 5 respectively.
Find : AB ,АС , BD , СО , AD and BC
Solution
АВ =|B-A|=| -3-(- 7)|3|] -3+7|=|4| = 4 units.
»AC =|C-A|=|0-(-7)|=|0+7|=|7|=7 units.
, مق =|D-B|=|5—(-3)|=|5+3|=|8|=8 units.
„CD =|D-C|=|5-0|=|5|= 5 units.
„AD =|D-=A|=|5-(-7)|=|5+ 7| =| 12| 12 units.
and BC =| C-B|=|0-(-3)|=|0+3|=|3|= 3 units.
From the following figure › complete :
x Y 2 Е E
ههه MÀ. э e+ +
-6 -5-4-3-2-10 12 3 4 5 O 7
[a] XY = vs [Ы] XZ = <...
[c] YE =. [d] XF = ——
[е] ZE = ——— [f] YF =~
(120)
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى Toss]
Unit Three
Graphing points in the coordinate plane
The position of any point in the coordinate plane is determined by
a unique ordered pair.
For example :
[1] To graph the point A (2 53) »
follow the following steps :
* Start at 0
* Move 2 units to the right.
* Then , move 3 units up.
[2] To graph the point B (- 4 »— 2) »
follow the following steps :
* Start at O
* Move 4 units to the left.
* Then , move 2 units down.
—
@ The horizontal line xx is called the "x-axis"
@ The vertical line yy is called the "y-axis"
@ The distance between two points in the coordinate plane _
To calculate the distance between two points in the coordinate plane د
do as follows :
[1] Determine the line segment joining between them.
[2] If it is parallel to OX (or хх) » calculate the distance as on a horizontal
ray (or a straight line) and if it is parallel to oY (or yy) >
calculate the distance as on a vertical ray (or a straight line).
otio 02) شی tonto
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى EE
(1)
For example :
[1] In the opposite figure :
(1) AB // OX
7 AB=B-A
= 6-1 = 5 units.
(2) AC // OY
ЛАС=С-А
= 4—1 = 3 units.
[2] In the opposite figure :
(1) AB JI xx
г. AB 2|B-A|2|]4- (- 3)
= |4 + 3|= 7 units.
(2) DC // уу
ЮС 2|C - D|2|3- (- 5)]
=|3 + 5|= 8 units.
[Example ©
In the coordinate plane :
[1] Determine the position of the following points : A (5 » – 1) , В (5 , 3),
C (7-253), D )- 2 ›— 1) and mention the name of the figure ABCD
[2] Find the area and the perimeter of the figure.
[3] Determine whether the shape is symmetric or not ?
©
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى EE
Unit Three
Solution
[1] The figure ABCD
is a rectangle.
[2]АВ -|B-A|*|3- (- 1|
=|3+1|=|4|= 4 units.
-. AB = CD = 4 units.
»DA=|A-—D|=|5—(-2)|
=|5+2|=|7|=7 units.
г. ОА = BC = 7 units.
2. The area of the figure = L x W = 7 x 4 = 28 square units.
»the perimeter of the figure = (L + W) x 2 = (7 + 4) x 2 = 22 units.
[3] The shape is symmetric.
Example (3
In the opposite coordinate plane :
ABCD is a rhombus.
[1] Find the coordinates of each of
the points A. B , C and D
[2] Find the area of
the rhombus ABCD
Solution |
[11A(1.-3).B(-2.2)
»С (1,7) and D (4 .2)
@
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى EE
[2] • The length of AC =|C -A|2|7 – (C 3)| 2|7 + 3|=| 10| = 10 units.
• The length of BD -|D - B| |4- (- 2)] 2]4 + 2|=|6|=6 units.
• The area of the rhombus ABCD = i x AC x BD
i x 10 x 6 7 30 square units.
In the coordinate plane » determine the position of each of the following
points : A (- 1 »—3) و B (4 ,- 3) and C (- 1 و 2( , then complete :
[а] AB = --------- units. و AC د units.
[b] The type of the triangle ABC with respect to its side lengths is ~-
the type of the triangle with respect to its angles is
[c] The area of A ABC = square units.
v
Now at all bookstores
e GE
in
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى EE
10
Distance between two points
in the coordinates plane
ÛÎ From the school book
(E) From the following figure, complete :
x Е F G H
و
Ф
-B -7 -6 -5 -4 -3 -2 =1 0 1 2
b دوع
Locate the points A (0 ,4) ,
B (2 , 1) and C )- 2 , 1) ; then
Find the length of BC (Giza 2017)
E] in the opposite coordinate plane :
4
@ Determine the position of the
following points : А (—3 .—3) ٠
В )-3 ,2) ,C )5 ١ 2) and D (5 ,– 3)
and mention the name
of the shape ABCD
Find the perimeter and the area
of the shape ABCD
(125)
ر هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى TERS
In the opposite coordinate plane :
a Determine the position of the
following points : A (- 4 »— 4)
> В (2 ,—-4) ,C (2.2) and
D (- 4 ,2) and mention the
name of the shape ABCD
b Find the perimeter and the area of
the shape ABCD
с Determine whether the shape is symmetric or not ? Why ?
Ш In the opposite coordinate plane :
a Determine the position of
the following points :
L(-1;1) MG of
N (1 »8) and E (- 1.8)
Find the perimeter and the
area of the shape LMNE
Determine whether the shape
is symmetric or not ? Why ?
129
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح اوله على مواقع أخرى EE
Unit Three
In the opposite coordinate plane :
( Determine the position of the
following points :
A(-15-4). B(- 1.3) and
С (5 »—4) » then find :
а The length of each of AB and AC
b The type of the triangle ABC
with respect to its sides and angles.
€ The area of A ABC
[7] n the opposite coordinate plane :
EFGH is a rectangle ; complete :
а The coordinates of the points
с The area of the rectangle = =- square units.
—
‘ar
EE هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى
!
( a £n the opposite coordinate plane :
ABCD is a rhombus.
a Complete the coordinates of the
following points :
(Suez 2015)
The area of the rhombus ABCD can be calculated by using the length
of its perpendicular diagonals › where :
The length of AC = =~
»the length of BD = -~-=
The area of the rhombus = --------:-
Ю in the opposite coordinate plane
ABCD is a square ; then complete :
3 The coordinates of the points :
b The length of AC
» the length of DB.
€ The area of the square ABCD = ----------
©
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى EE
EO WES SOR Oy
œ
a
GNO Tl «C7
Sa C =
SSS!
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح ан ш: را p I
н] f سمس M
You have known that , there are three types of the geometric %
transformations which are shown іп the following diagram : 8
a nii Li \'
Reflection - C Translation h Rotation 2
ROA
R|A te | (RS
\
You have studied last year "The reflection" and a general idea about
geometric transformations.
Now ; you will study with more details "
(QV in Y ايتداتى / تيرم ١/ رياضيات لفات wall (129)
R
4
&
M
4n
»
7
Eo!
| Prelude
If a car moved in a straight line a distance
of 25 metres forward. In this case , we
can describe the movement of the car as
a translation of 25 metres forward.
This movement changed the place of the
car without changing its orientation.
i.e. To determine the new position of the car after its movement د
we should know two important elements which are :
The magnitude of the translation (25 metres).
The direction of the translation (forward in a straight line).
Definition : The translation is a geometric transformation which slides a shape
from а place to another place (image) such as every point of the original shape
moves the вате distance in the same direction to form the image.
• To find A which is the image of A
by translation MN in the direction
of MN › we do as follows :
STEP 1
Draw from A a ray parallel to MN and in the сате direction.
STEP 9
By the compasses in A ae a centre with radius length = MN >
draw an are to intersect the ray drawn from A at the point A
(AA = MN and AA // MN)
А is the image of A by translation of
magnitude MN in the direction of MN
тел: |
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى [сте
Unit Three
l Example (1)
Find the image of the point A by
translation of magnitude 3 cm.
in the direction of MN.
Solution
[1] Draw from A a ray parallel to MN
and in the same direction of MN
[2] By the compasses in A as a centre
with radius length = 3 cm. , draw an arc to
intersect the ray drawn from A at the point А
s the image of A by translation of
magnitude 3 cm. in the direction of MN
* To find the image of AB by translation
MN in the direction of MN › we do as
follows :
Find the image of the Similarly › ше find the
point A by translation MN : image of the point B by
in the direction of MN ae : translation MN in the
we mentioned before , say A | direction of MN » say B.
Oraw AB to be the image
of AB by translation MN
in the direction of MN
Check that : AB = AB and AB // AB
E)
EE هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى
» To find the image of a geometric shape by a given translation,
translate each vertex by the same given translation as shown
before in (A).
The opposite figure shows the image of
a triangle by a certain translation.
Every point of the shape must move :
• The same distance.
* In the same direction.
ШЫ acce
Using the geometric tools, draw the image of
Original shape
АВ by translation MN in the direction of MN
Second Translation in the s plane
In the opposite graph :
If A ( Ell - El) is a point in the orthogonal
coordinates plane and to find its image A
by translation with magnitude El length
units in the direction of OX followed by
a translation with magnitude Ё length
units in the direction of OY , we get A to
be the point (5 » 5)
ie A( B+ E: +B)
)32
Й هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى [ЖЕЕ]
Unit Three
According to this :
f The image of the point A (x y) BY translation (a >b) the point À (x+ a sy + "|
i.e. Translation in the orthogonal coordinates plane transforme each point A in the
plane into ite image А in the same plane by a displacement (a) in the direction
of the x-axis followed by a displacement ei in the direction of the y-axi
| Example ©
Find the image of the line segment AB where A (- 4 » 3) , B (2, 0) by
the translation (X » y) حب (X + 2» y — 3)
Solution
'. (X ¥) (xX + 2 ,у-– 3) , then :
٠ The image of A (- 4 , 3)
БА (-4+2,3-3)
ie. Ã2 0)
• The image of B (2 , 0)
6 В (2+2,0-3)
їе. В (4 ,— 3)
The translation (x » y) — (x + 2 » y — 3) transforms each point to
another point by a horizontal displacement of 2 units to the right and
a vertical displacement of 3 units downwards.
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى EE
tO
Example (3 )
Draw оп a square lattice A ABC where A (4 » 4) و B (0 » 2) and C (6 »— 2)
و then find its image by the translation (X »y) هب (X- 4 »y + 1)
Solution
Its image by
the translation
(Х ›у) (x-4»y*1)
A(4 54) А (0,5)
в(0›2) B(-4 3)
C(65-2) @(2,-1)
The point
^ ABC is the image of A ABC by
the translation (X , y) —» (X —4 »y + 1)
i L|
E3 4 f
-ABNA > Hê Bê and АС/АС | |
-АВ=АВ „ BC=BC and AC=AC | ||
| ل nB) =m (ZB). and mcs m حر
The translation (X » y) ——» (X + a › y + b) can be written simply :
"The translation (a » b)"
For example :
The translation (X و y) جه (X + 2 » y — 1) can be written simply :
"The translation (2 »— 1)"
On a square lattice , draw A ABC where A (- 3 , 2) ; B (- 1 » 1) and
С (- 2 , 0) , then find its image by the translation :
(Ху) ——» (X +2 sy +1)
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى EE
Unit Three
Example (4 )
On a square lattice و draw the rectangle ABCD where A (4 » 3) و B (4 » 1)
و 1( © و 1) and D (1 » 3) و then find its image by the translation (- 2 »— 3)
Solution
The point
Its image by the
translation (- 2 .— 3)
A (4 ,3) А (2,0)
В (4,1) В (2 5-2)
C(t») CC ED
D (153) D (-150)
2. The rectangle ABCD is the image of
the rectangle ABCD by translation (- 2 , – 3)
Example ( 5)
If the image of the point A (— 3 و 2) by translation in the Cartesian
coordinates plane is A (2 »— 2) :
[1] Find the rule of translation.
[2] Find the image of B (1 , – 3) by the same translation.
Solution |
[1] By noticing the opposite graph
» we find that the translation which
makes A (2 ,— 2) the image of
А (- 3.2) is equivalent to :
* Horizontal displacement of 5 units
to the right (5)
* Vertical displacement of 4 units
downwards (— 4)
-. The rule of translation is
(x :y)— (X +5 ,у-– 4)
[2] B(1,-3) جه B (14 5.—3—4)
i.e. B (6-7)
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى EE
Example (6)
If A (7 »— 2) is the image of A by the translation
whose rule is (X » y) (X-— 3 ,y + 1) »find A
Solution
Let A be (X , у) q
“A(X sy) — À (х—3»у+ 1) G2emember that :
STA 2) (ху) = (a ob) >
s(x-3.y*1)2(75.-2) thenx=a>,y=b
1 х-3=7
»у+1=-2
ЛА (10 ,-3)
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى [сту
Geometric transformations (Translation)
EQ From the school book
[1] In the following shapes determine the type of the geometric transformation
(reflection , translation or rotation) :
` FF cE ЯЕ
T
|
h
Determine which of the following shapes is symmetric and which is not
symmetric د then draw the axes of symmetry :
a | b | e
W
OAL Y رياضيات لغات /3 ابتدائى / تيرم walsell аз?
ر هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى Teese]
©)
V
Complete the following table :
The point The translation The image
(253) (X5»y)—-(X*35y*1)
— M" (х,у) —+ (Х+2,у- 1)
(0 ,- 3) (X s y)— (X+ yt
EAEN (Xy) — (X+3 sy +1)
E complete each of the following :
a Two elements must be known for the translation :
(217-23
b The image of the point (2 , 5) by translation
(х,у) جح (x *2.y*1)is
The image of the point (— 5 , 4) by translation
(Ху) —» (Х+4,у- 5) іѕ
The image of the point (3 , 2) by translation
(X*35y-2)is- . (Giza 2013)
The image of the point A (— 5 , 2) by translation
(-15-3)isA( (El-Sharkia 2011)
The image of the point (2 , — 1) by translation
(-355)is (Assiut 2016)
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى EE
Unit Three
The image of the point A (4 ; 5) by translation
The image of the point A (2 , — 1) by translation
(x-15y*3)is --------- (Suez 2016)
The image of the point (0 , 2) by translation
(X*15y*3)is( (Alexandria 2011)
The image of the point (— 2 . — 5) by translation
(X5 y) حب (x-2;y)is =
The image of the point (3 , — 2) by translation
(х,у) —+ (х,у *3)is
The image of the point ·-------- by translation
(X sy) ——» (X-2 sy + 3) is (7 , 4)
The point (a , b), its image is (5 ›— 4) by translation (2 , – 3) , then the
coordinates of the point (a » b) = ===- (El-Kalyoubia 2017)
If the image of the point (3 › 2) is the point (6 , 1) , then the translation
rule is (X و y) جه (5
The image of the point A (3 , 6) by translation 3 units in the negative
direction of X-axis is -
If À is the image of A by translation of magnitude (MN) in the direction
of MN » then AA = <... (El-Monofia 2011)
In the opposite figure :
А is the image of A by translation.
A
Яст,
A
Its magnitude is 7 uiii RN
N
in the direction of -——- (Red Sea 2011)
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى EE
5
Choose the correct answer :
( а The image of the point A (1 , 2) by translation (1 ,— 1) is
(а) (2 > 1) (b) (2 » 3) (с) (1-1) (9) (1,3)
b £ The image of the point A (— 4 , 3) by translation (– 1 ,— 4)
i (Damietta 2017)
(а) (735-1) (b)(-7 ,3) (с) (-5 »-7) (9) (-5 »-1)
The image of the point (3 , — 2) by translation (4 » 2)
i (El-Kalyoubia 2015)
(а) (-7 »0) (b) (7 »0) (с) (- 1.4) (d) (1,7)
The image of the point (3 , — 2) by translation (— 3 , 2)
i (El-Beheira 2016)
(a) (0 » 0) (b) (2 » 0) (с) (3 0) (9) (6,4)
The image of the point (4 , — 3) by translation (— 2 , 3)
is (Ismailia 2012)
(a) 2 »0) (b) (6 ,—6) (с) (-6 ‹6) (d) C2. -1)
If (X › y) is the image of the point (3 , — 2) by translation (1 , 3)
»then the point (X 5 y) = з
(a) (2 51) (b) (2 4) (с) (1 »4) (d) (4,1)
The image of the point A (3 , – 4) by translation (X + 1 » y + 4)
i (Giza 2012)
(a) (4 ,0) (b) (4.8) (с) (3 » 8) (d) (3,0)
The image of the point A (5 ; 1) by translation (X — 1 sy — 1)
i (El-Kalyoubia 2011)
(а) (4 +0) (b) (6 +2) (c) (4.2) (9) (- 4 5-2)
If A (1 , 2) , then the image of A by translation (X + 1 ¥ — 1)
(Suez 2013)
(а) (2 » 3) (b) (2,1) (с) (1,1) (d) (1,3)
CA The image of the point (3 , 5) by the translation (X + 2 , y — 1) is
(a) (6,6) (b) (5 > 4) (с) (1 »4) (9) (1,6)
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى EE
Unit Three
К & The image of the point (- 1 , 2) by translation of magnitude of З units
in the positive direction of the x-axis is
(а) (1 , 5) (b) (2.2) (с) (2,2) (d) (- 1.3)
| The image of the point (— 3 , 4) by translation of magnitude of 4 units
in the negative direction of the y-axis is (El-Beheira 2015)
(а) (- 3.0) (b) (- 7 »4) (с) (- 3.8) (9) (- 14)
m The image of the point (3 ; 0) by translation of magnitude 3 units in the
negative direction of X-axis is ·--------.
(a) (0 0) (b) (3 » 3) (с) (3 ,– 3) (d) (0 ,– 3)
n The image of the point (2 , — 1) by translation of magnitude 3 units in
the positive direction of y-axis is
(а) 2:2) (b) (5 :-1( (с) (5 2) (d) (2-4)
© If A (3 , — 3) is the image of A by translation (х,у) —+ (xX-1,y-4),
then the point A is
(а) (2 ,– 7) (b) (4 , 1) (с) (4 ,– 1) (d) (2,1)
Find the image of each of the following figures by the indicated translation :
(х,у) —+ (х-3.у- 4) (х,у) —+ )<+ 2 ,у +3)
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى EE
by translation (3 , – 4)
translate 1 unit to the right
and 4 units downwards.
а the opposite figure :
Find the image of the line segment
AB where A (1 , 2), B (-2 .— 1) by
translation (X + 2 , y — 2)
(South Sinai 2011)
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى EE
Unit Three
B EQ In the opposite coordinate plane :
( a Determine the image of DE where
D (2 : 0) and E (- 1 , 1) by translation
(х,у) —+(X+3,y +2)
b What is the name of the shape
DDEE ? Why ?
ABCD is a rectangle where A (4 ; 3) ,
B(4.1).C(1.1)andD (1,3)
Find its image by
translation (X — 2 » y — 5)
a Determine the coordinates of
the following points :
Find the image of the parallelogram
ABCD by translation
(х,у) حب (X *2»y-4)
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى EE
(2)
( (ED) С^ ın tne opposite figure :
a Determine the coordinates
of the following points :
Find the image of the A ABC by
translation (X ; y) مت (X +2 sy + 3)
The length of BC
The length of AB =
Is A ABC symmetric or not ? Why ?
n the opposite coordinate plane :
7
Determine the following points :
A(25;-2).B(1;1)andC(15,6) |
Find À which is the image of the
point A by translation (2 , – 1)
Find BC which is the image
of BC by translation (3 » 0)
Find BC and BB
Calculate the perimeter and
the area for the shape BBCC
(B ما لع the opposite figure :
Find the image of the point A by
the translation ME
in the direction of MN
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى EE
Unit Three
Using the geometric tools , draw the image of each of the following by
translation MN in the direction of MN as shown in each case :
M А M N
| A
Fig. (1) Fig. (2)
Using the grid , draw the image of LI
the figure ABCD by the translation |
of 4 units in the direction of BC -
bah LIC
[16] Determine in the coordinates plane the image of the line segment AB
where A (2 , 3) and B (- 2 ; 0) by translation (X + 3 لاو — 2) (Alexandria 2012)
In the coordinates plane :
Draw the А ABC where A (0 , 1) ,В (2 , 3) and C (- 1 , 4) » then find its
image by translation (X + 2 لاو + 3) (Damietta 2013)
Ba Draw А ABC where A (1 , 1) د B (- 3 ,– 1) and C (0 , —5) ,then
determine graphically its image by translation (5 » 0)
Plot the points O (0 , 0) B (2 , 0) and C (0 , 2) on the coordinate plane ,
then find :
a The length of OB
b The image of the triangle OBC by the translation (x + 2 ,у + 2)
(Cairo 2015)
EB Determine in the coordinate plane the following points A (- 3 , 4) ,
B (1 ,4) and C (1 , 2) , then find:
a AB = .......... ВС
b The image of A ABC by the translation (0 , — 3) (Et-Sharkia 2017)
QAO) Y ابتدائى / تيرم V رياضيات لغات vals
EE هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى
( O Represent the points A (2 , 3) B (4 +3) and C (4 , 7) in the lattice 5
then find :
a BC- ~ length unit د AB = --------- length unit
b The image of A ABC by translation (0 »— 4) (Port Said 2016)
€ The area of AABC
Determine in the coordinates plane the rectangle ABCD image where
A (2 +3) >B (2 » 1) >C (-2 » 1) , D )- 2 »3) by translation (X + 3 لاه + 3)
What is the type of the resulting figure ABCD ? (El-Fayoum 2011)
n the coordinates plane :
Draw the rectangle ABCD where A (4 , 2) »B (4 ,4) C (1 ,4) and D (1 » 2)
a Draw its image by translation (X + 2 ,у + 2)
b Calculate the perimeter of the image of rectangle ABCD (Cairo 2013)
(El Fina the image of each of the following points by the translation
(х,у) جب (x + 2 › y — 3) followed by the translation
(х,у) حب (х-3 ,у +1)
а (4,-2) b (1,3) с (0,2)
|25 Use the translation (X و у) —— (X + 2 ,у + 3) to locate the point whose
image is (2 :3(
£A The image of (a ; b) by translation (2 »— 3) is (5 »— 4) » find (a » b)
(Matrouh 2017)
If the image of the point A (1 » 1) by translation in the plane is А (2.2)
» find the images of the points О (0 , 0) , B (- 1 » 3) and C (—3 , 5) by the
same translation.
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى Tus]
Unit Three
(
If A (- 3 , 1) and B (1 »— 2) , write the rule of the translation that makes
B the image of A
The point A (3 , — 3) is the image of the point A by the translation
(X » y) حب (X - 1 » y — 4) Locate A , then by the same translation ›
draw the image of A ABC where B (5 , 0) and C (- 1 ,— 2)
In the opposite figure ;
Copy the graph , then draw the triangle ABC
whose image is A ABC by the translation
(х,у) —- (x *2.y +3)
State whether the graph shows a reflection and name the line of reflection
oratranslation and describe the translation :
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى Tues]
WWW:zakrooly:com
N
3 Area of the circle
N
0 You have studied before what is a circle, and what we mean by a circular sector. 4
P You have known that : :
{ A circular sector is a part of a surface of a circle bounded by an are and two | | 9
| radii passing through the ends of the arc. | X
x ы 1 b
For example :
The coloured part in the opposite circle
represents the circular sector AMB
Finding the area of j n
• The figure below shows the surface of a circle divided into equal circular [^ ١
sectors. These sectors were arranged in a certain way to form a shape E
similar to a parallelogram.
\
-ANW R
9
4 |
&
کا ما خاس برق اکر تعب دس تيه ل برق نر اك
. | الصف السادس الابتداثى | M LII үк
Unit Three
* If the number of circular sectors increases; then the shape will be closer to
a rectangle.
C1 The width of the shape is the radius length
The length of
the shape is half
the circumference
3 7-Half circumference (л x r)—
Circumference
The area of the circle = the area of the resulted rectangle
= Length x width
= half the circumference x radius length
= 1(2лг)ухг=лгхг=лг?
Where
So, The area of the circle - r? m= 22 23.14
| Example (1)
Find the area of a circle whose radius length is 14 cm. (Consider т = 22)
Solution
A= 2x7? = 22 x (14)? = 616 cm?
Example ( 2 )
Find the area of each of the following circles (Consider 1 = 3.14) :
э t С
Fig. (1) Fig. (2) Fig. (3)
Solution
Fig. (1): A= x r? = 3.14 x 52 = 78.5 cm?
Fig. (2):А = xt r? = 3.14 x 62 = 113.04 m?
Fig. (3): d= 15 cm.
те »ا } دل ل 15 - 7.5 ст.
г. A =T r? = 3.14 x (7.5)? = 176.625 cm?
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى FEE]
| Example (3
In the opposite figure за circle M of a radius length
14 cm. » is divided into eight equal circular sectors.
Calculate the surface area of one sector.
(Consider x = 22)
Solution
ب. A= 2 = 22 x (14)? = 616 cm?
^, The area of one sector = 616 + 8 = 77 cm?
[Example (4)
In the opposite figure » find the area of
the coloured part. (Consider X = 3.14)
Solution |
‘~ The area of the greater circle = л r? = 3.14 x (7.5)?
= 176.625 cm?
„the area of the smaller circle = zt r? = 3.14 x (4.5)?
= 63.585 cm?
2. The area of the coloured part = 176.625 — 63.585 = 113.04 cm?
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى EE
Unit Three
)
In the opposite figure و find the area of
the coloured part. (Consider I = 3.14)
Solution
*- The area of the square = 20 x 20 = 400 cm?
»the area of the circle = zt г? = 3.14 х 10? = 314 cm?
-. The area of the coloured part = 400 — 314 = 86 cm?
Example (6
Calculate the area of the opposite figure.
(Consider 7 = 22)
| Solution 10 cm.
* The area of the rectangle = L x W = 10 x 7 = 70 cm?
» the area of the semicircle = 4 Tes i x 22 x (3.5)? = 19.25 cm?
7. The area of the whole figure = 70 + 19.25 = 89.25 cm?
"E acce
In the opposite figure :
ABCD is a rectangle its length is 14 cm.
and its width is 10 cm. A circle is drawn
to touch the sides AB and CD
Calculate the area of the coloured part.
(Consider 7t ع 3.14)
Example (7
Calculate the area of the circle whose circumference is 44 cm.
(Consider л = 22 )
Solution
7 The circumference = 2 tr
.. د جه 2, 22 xr 445 44 xe
7 7
44х7 ш -A= = 22 2 = 2
T ك 7 от. A= 7 х 72 = 154 cm?
ык
С)
EE هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى
Area of the circle
LI From the school book
(GB Find the area of each of the following circles (Consider = 3.14)
b сш
Area = eee
(Suez 2014)
Area = oo
Area = с Area = с
EJ Find the area of each of the following circles for the given radius.
Round your answer to the nearest hundredth (Consider 7t = 3.14) :
а г=8ст. | b r=3.6m. с г= 5 кт.
d г= 14 т. е г= 6.3 тт. f r2 78.4 dm.
Find the area of each of the following circles for the given diameter.
Round your answer to the nearest hundredth (Consider 7 = 3.14) :
b d-21m. | с d=1.4cm.
е d-18.8dm. | f а= 28km.
а а= 16cm.
d d=40 mm.
(Souhag 2015)
Calculate the area of a circle of radius length 7 cm. (Consider Jt = 22 )
)82(
ر هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى FEES
Unit Three
(B^ circle with radius length 4 cm. Calculate its area. (Consider л = 3.14)
(Luxor 2011)
Ш A circle › its diameter is 12 cm. Calculate its area.
(Consider 7 = 22 or 3.14) (Et-Sharkia 2016)
Ga circle د its diameter length is 20 cm. Find its surface area.
(Consider 7 = 3.14) (Souhag 2017)
[ 5] Find the area of a circle with diameter of length 17.5 cm. (Consider л = 22)
D^ circle › its diameter is 14 cm. Calculate its surface area and its
circumference. (Consider 7t = 22 ) (El-Monofia 2015)
E C2 In the opposite figure :
A circle M of radius 4 cm. ; is divided into
five equal circular sectors.
Calculate the surface area of one sector. (Consider zt = 3.14)
[11] In the opposite figure :
A circle M of radius length 7 cm. ; is divided into
eight equal circular sectors.
Calculate the area of one sector. (Consider 7t -22)
(Matrouh 2015)
QJ A circular birthday tart the diameter of its
upper base equals 25 cm. is divided into eight
equal circular sectors , then find the area of
one sector "Approximating the result to the
nearest integer"
(Consider л = 22 or 3.14)
wall (53 رياضيات لغات /\ ابتدائى / تيرم Qr D Y
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى EE
3
Lesson
Find the area of each of the following figures (Consider л = 22) :
(а) ш (bo ca о
M
«Souhag 2014»
10.5 cm.
5 In the opposite figure :
Circle M is drawn inside a square of side length 14 cm.
and touches its sides.
Calculate the area of the coloured part.
(Consider xt = 22)
In the opposite figure :
ABCD is a rectangle its length 12 cm. and its
width 7 cm. Calculate the area of the coloured
part. (Consider 7 = 22) «El-Kalyoubia 2017»
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى EE
Unit Three
[16| In the opposite figure :
A rectangle where its length = 34 cm. and its
width = 15 cm. , and a semicircle of centre M
Calculate the area of the coloured part.
(Consider Л = 22 ) “Red Sea 2013» 34 cm.
Find the area of the coloured part of each of the following figures
(Consider Jt = 3.14) :
(c)
«Ismailia 2011»
( Complete :
(a)The surface area of the circle = <... «Cairo 2012 - Matrouh 2015»
(b) The radius length of a circle is 14 cm. , then its circumference = --------- ст.
and its area = 0. cm? (Consider л = 22)
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتد اوله على مواقع أخرى ]1525
The diameter of a circle is 20 cm. , then its circumference = =
and its area = ست cm? (Consider x = 3.14)
The surface area of the circle of radius length 7 cm. = + zt cm?
(Beni Suef 2016)
A circle, its area is 25 л cm? , then the length of its radius is
(Damietta 2016)
If the circumference of a circle is 30 л mm. , then the area of this circle
equals =-=
m. — 12 م
The opposite figure is formed of a quarter of
a circle and a triangle ; then its area = — cm? En C
(Consider 7 = 3.14)
The area of the opposite figure
= ст? (Consider zt = 22)
The area of the coloured part
cm? (Consider zt = 22)
The area of the coloured part
mee cm? (Consider zt = 22 )
k The area of the opposite figure
= = om? (Consider 7 = 3.14)
Choose the correct answer :
а EQ The area of a circle = ------- (Qena 2017)
(лг or xr? or 2xr or 2712)
b The circumference of a circle = ---------
(лг or 2r or xi? or 2712)
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى EE
Unit Three
A circle, its radius length is 3.5 cm. , then the surface area = - cm?
(Consider x = 22) (Et-sharkia 2012) (11 or 22 or 38.5 or 384 )
A circie with radius length = 1 cm. ; then its area = cm?
(ismaitia 2011) (Ж or 276 or $n or x?)
The area of the circle whose diameter length is 8 cm. "TX cm?
(E-Menia2015) (4 or 8 or 16 or 64)
A circle › its diameter length is 6 cm. , then its surface area = ·......... cm?
(üsmaiia201242 ) 3 IU or 6л or 9л or 367)
The circumference of a circle is 44 cm. ; then the length of its diameter
is ست cm. (Consider л = 22 ) (14 ог 22 ог 44 ог 154)
The perimeter of the opposite figure = --------- cm. (Souhag 2017) 1
(2л or 5л or 2+4 or 4n+4)
The area of the opposite figure = .........٠: cm?
(16% or 47 or 27 or 472)
The area of the coloured part = ст?
where r = 2 cm.
(2-% or 2-2 or 2-25 or 2+ 27(
(Ba circle its circumference is 14 л m. Calculate its area.
йл circle its circumference is 2 zt cm. Calculate its area.
£ A circle its circumference is 88 cm. Find its radius length and its area.
(Consider л = 22 ) (Alexandria 2015)
( ©2 A circle its circumference is 62.8 cm. Calculate its area.
(Consider 7 = 3.14) (Е(-Веһеіга 2012)
LJ A circle its circumference is 44 cm. Calculate its area.(Consider л = 22)
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى Tues]
KS)
£3 In the opposite figure :
Find the area of the coloured part. (Consider Jt = 3.14)
B Ш A table its surface in the form of a circle , its
/ diameter is 1.5 m. Its surface is wanted to be
covered by a sheet of glass equal to its surface.
Calculate the cost price if the square meter
of the glass costs L.E. 60 (Consider t = 2? or 3.14)
If the length of the outer diameter of a computer CD is
12 ст. › and the length of the inner diameter is 1.5 cm.
Find the area of this CD. (Consider zt = 3.14)
(Б) A garden » whichis circular in shape sits
circumference is 132 metres »find :
a The length of diameter of the garden in metre.
b The area of the garden in square metre.
(Consider 7t = 22)
E C2 In the opposite figure :
Mis a circle of radius length 5 cm. » a rectangle is
drawn inside it , its length is 8 cm. and width is 6 cm.
Calculate the area of the shaded part.
(Consider л = 22 or 3.14)
Calculate the coloured area of the opposite figure if
the radius of the large circle is 6 cm. and the radius
of each of the small circles is 2 cm. (Consider 7 = 3.14)
For Excellent Pupils
a circle its area is 616 cm? Calculate its radius length and its circumference.
(Consider 7t = 22)
|
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى [сте
жа 4
) 1
| Definitior
Я | • The lateral area of a solid is the eum of the areas of all ite faces that are not bases. ў
(e The total area of а solid is ће вит of the areae of all ite faces included the bases. \'
Rear acu au »4
كه @
[emember that: ——— — — 6 faces | ده uu — ]
| |
8 vert ! |
@ The cube has б faces › each face is нн | } n
2 a square and all faces are equal in area. 2
So , the area of one face = the area of the square + 2 n
O 3 Length 55 4
I9 4 = the edge length x itself !
(©) The cube has 8 vertices and b \
@ The cube has 3 dimensions : “length » width and height" and they are 4
equal in length. ^»
© The volume of the cube = The edge length x Itself x Itself $
= J
: D
ez 159 №
هذا as خاس يموقع cll ly Sal وا يسع بتداوله ШЖ ==
sated! oie, كمع «йу 5 2525)
If we have a carton box in the form of a cube, and we want to see how to
find the lateral area and the total area of a cube, we can unfold the box as
shown in the figure below :
unfolded
—-
• The lateral area of a cube is the sum of the areas of its four lateral faces
(1) » (2) › (3) and (4) which are perpendicular to the base of the cube.
So, The lateral area of a cube = The area of one face x 4
7 Edge length x Itself x 4
When the faces of the cube were unfolded »
the rectangle ABCD was formed from the
lateral faces.
* The length of this rectangle
= The sum of the edge lengths of the four
lateral faces (1) » (2) › (3) and (4) which
represents the perimeter of the base of
the cube.
FO
* The width of this rectangle = The height of the cube
So, The lateral area of the cube = Perimeter of the base x Height
* The total area of a cube is the sum of the areas of all the faces of the cube.
So, The total area of a cube = The area of one face x 6
Edge length x Itself x 6 =
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى Tus]
Unit Three
The lateral area
4
The total area
@ The area of one face = pex = t
@ The area of one face = = 1 x (the lateral area)
x (the total area)
[Example (1)
A cube-shaped Бох » whose edge length is 3 cm. Find :
[1] The lateral area.
[2] The total area.
Solution
The area of one face = 3 x 3 = 9 cm?
[1] The lateral area = the area of one face x 4 = 9 x 4 = 36 cm?
[2] The total area = the area of one face x 6 = 9 x 6 = 54 cm?
Example (9)
The lateral area of a cube is 28 cm? Find its total area.
Solution
The area of one face
3 2
4 7 cm:
The total area = the area of one face x 6 = 7 x 6 = 42 2
= the lateral area _ 2
4
| Example (3)
If the perimeter of a face of a cube is 20 cm.
Find its lateral area and its total area.
Solution
The side length of a face = the perimeter + 4 = 20 + 4 = 5 cm.
The area of one face = 5 x 5 = 25 cm?
The lateral area = 25 x 4 = 100 cm?
The total area = 25 x 6 = 150 cm?
Qr 0 Y ابتدائى / تيرم а رياضيات wall
EE هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى
If the sum of the edge lengths of a cube is 108 cm. ; find :
[1] The lateral area.
[2] The total area.
Solution
The edge length of a cube = 108 + 12 = 9 cm.
The area of one face = 9 x 9 = 81 cm?
[1] The lateral area = 81 x 4 = 324 cm?
[2] The total area = 81 x 6 = 486 cm?
[1] A cube of edge length 7 cm.
Find its lateral area and its total area.
[2] If the sum of the edge lengths of a cube is 96 cm.
Find its total area.
a and total area of a cuboid
fuus that : — ————— 6 faces
B vertices
@ The cuboid has 6 faces د 12 edges
each face is a rectangle and each
two opposite faces are equal in area. Length
So, the area of one face = The area of the rectangle = Length x Width
©@ The cuboid has 8 vertices and |2 edges.
@ The cuboid has 3 dimensions : "Length , Width and Height"
© The volume of the cuboid = Length x Width x Height
= Base area x Height
EE ر هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى
Unit Three
If we have a carton box in the form of a cuboid, and we want to see how to
find the lateral area and the total area of a cuboid. we can unfold the box as
shown in the figure below :
H
G
EE. Unfolded Haight i
A B A
E
• The lateral area of a cuboid is the sum of the areas of its four lateral faces
which are perpendicular to the base of the cuboid.
i.e. The lateral area of the cuboid (in the diagram) whose base is
the rectangle ABCD
= The area of ABFE + The area of BCGF + The area of CDHG
* The area of DAEH
=ABxh+BCxh+CDxh+DAxh
= (АВ + BC + CD + DA) xh
= The perimeter of the base x The height
So, The lateral area of the cuboid = The perimeter of the base x The height
and we deduce that :
8 _ The lateral area of the cuboid
The бе опа rr. perimeter of the base
The lateral area of the cuboid
* The perimeter of the base = Tha height
<5 Ж
EE هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى
• The total area of the cuboid is the sum of its lateral area and the areas of its
two parallel bases.
So, The total area of the cuboid = The lateral area + 2 x (The area of the base)
If a cuboid without a lid » then :
the total area = The lateral area + The area of one base
| Example (5
A cuboid-shaped box is 6 cm. long »
4 cm. wide and 7 cm. high 5 find :
[1] The lateral area.
[2] The total area.
Solution | ват
[1] The perimeter of the base = (L + W) х 2 = (6 + 4) х 2 = 20 cm.
The lateral area = The perimeter of the base x The height
= 20 x 7 = 140 cm?
[2] The area of the base = 6 x 4 = 24 cm?
The total area = The lateral area + 2 x (The area of the base)
= 140 + 2 x 24 = 188 cm?
The inner dimensions of a swimming
pool are 24 m. » 16 m. and 3.5 m. ; it is
necessary to cover its inner floor and
sides with square-shaped ceramic
tiles of side length 20 cm.
How many tiles are needed ?
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتد اوله على مواقع أخرى ]1525
Unit Three
Solution
The perimeter of the base = (L + W) x 2 = (24 + 16) x 2 = 80 m.
The lateral area = the perimeter of the base x the height
= 80 x 3.5 = 280 m?
The area of the base = L x W = 24 х 16 = 384 m?
Since , the swimming pool without a lid د
Then د the total area = the lateral area + the area of the base
= 280 + 384 = 664 m? = 6 640 000 cm?
The area of one tile = 20 x 20 = 400 cm?
total area of swimming pool
the area of one tile
= $620 000 - 16 600 tiles.
Then the number of tiles =
Example (7
A room has a square floor of side
length 4 m. and height 2.8 m.
It has a door of width 90 cm. and
height 2.2 m. and two windows
each of dimensions 1 m. and 60 cm.
The walls and ceiling of the room
are painted. If the cost of one square
metre to be painted is P.T. 475
Find the cost of painting the room و
approximated to the nearest pound.
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى EE
Solution |
The perimeter of the base = 4 x 4 = 16 m.
The lateral area = 16 x 2.8 = 44.8 m?
The area of the ceiling = 4 x 4 = 16 m?
The total area = 44.8 + 16 = 60.8 m?
The area of the door = 0.9 x 2.2 = 1.98 m?
The area of the two windows = 2 (1 x 0.6) = 1.2 m?
The total area of the painted part of the room - 60.8 — (1.98 * 1.2)
= 57.62 m?
The cost = 57.62 x 4.75 = 273.695 =L.E. 274
[1] A cuboid of length 8 cm. , width 6 cm. and height 7 cm.
Find its lateral and total area.
[2] A cuboid-shaped box with a square base whose side length
is 5 cm. and its height is 10 cm.
Calculate the lateral area and total area.
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى EE
Lateral area and total area for
each of the cube and the cuboid
(GB Calculate the lateral area and total area for each of the following cubes :
а m
©) From the school book
15 cm.
A cube of edge length 6 cm.
Find its lateral area and its total area. (El-Fayoum 2017)
A cube of edge length 8 cm. Calculate its lateral area and its total area.
If the edge length of a cube is 1.5 cm. Find its total area.
Find the total area of a cube whose face area is 49 cm?
(2 If the lateral area of a cube is 36 cm? Find its total area. (Assiut 2012)
a nd the lateral area of a cube whose total area is 48 m?
B The sum of the edge lengths of a cube is 48 cm. Find :
a The edge length of the cube. b Its lateral area.
© Its total area. (Beni Suef 2013)
The sum of edge lengths of a cube is 84 cm. Find :
a Its lateral area.
b Its total area. (El-Gharbia 2014)
ر هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى Tenes]
The sum of edge lengths of a cube is 120 cm. Find :
a The lateral area of the cube.
b The total area of the cube.
© The volume of the cube. (Cairo 2013)
(EI) The perimeter of one face of a cube is 24 cm. Find :
a The edge length of the cube.
b The lateral area of the cube.
€ The total area of the cube.
Ш The perimeter of the base of a cube is 28 cm.
Calculate its lateral area and total area.
KE] The total area of a cube is 384 cm? Find :
a Area of one face of this cube.
b Its edge length.
€ Its lateral area. (El-Dakahlia 2011)
LAM
[Bir the total area of a cube is 216 m? Find its lateral area and its volume.
LS _ للح سم
CJA cube is of edge length 8 cm. Calculate the ratio between its lateral
area and its total area. (El-Gharbia 2017)
CA When folding the opposite shape :
a The formed solid is
b The lateral area of this solid is
© The total area of this solid is
@ Complete the following table considering the unit length measured by cm. :
Cube Edge length | Lateral area | Total area
8
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى EE
Unit Three
( B Complete :
а The lateral area of a cube = ~- ы (Luxor 2012)
The total area of a cube = ·--------- (Kafr El-Sheikh 2014)
The edge length of a cube = 5 cm.
»then the area of one of its faces = =-
A cube of edge length 6 cm. ; then its lateral area = =-=- cm?
(El-Gharbia 2014)
The total area of a cube of edge length 4 cm. = .......... cm?
(Cairo 2015)
The edge length of a cube is 50 mm. ; then its total area is
The base area of a cube is 36 cm? , then its lateral area is =-=-
If the area of one face of a cube = 5 cm? , then the total area
of this cube = =- cm? (Alexandria 2011)
The sum of the edge lengths of a cube equals 72 cm., then the length
of the edge equals =- cm.
The sum of the edge lengths of a cube = 24 cm., then the area of one
face = ст? (Assiut 2016)
If the perimeter of one face of a cube = 12 cm.,
then its total area = ~-= cm? (Et-Menia 2016)
The face area of a cube is 4 cm? , then its volume = <... cm?
(El-Monofia 2016)
The lateral surface area of a cube is 100 cm? , then its volume
equals 5 (Cairo 2012)
If the volume of a cube is 1000 cm? , then its total area = cm?
(El-Dakahlia 2015)
The ratio between the area of one face of a cube and its lateral area =
The ratio between the area of one face of a cube and its total area =
The ratio between the lateral area and the total area of a cube = —
If the ratio between the edge lengths of three cubes is 1:2:3,
then the ratio between their lateral areas = ---------- E
TERRE
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى EE
(4)
Choose the correct answer from the given ones :
а The lateral area of the cube = Area of one face x -= (south Sinai 2013)
(а) 2 (b) 4 (c) 6 (d) 8
b Асире of side length 4 cm. , then its lateral area = . cm? (Beni Suef 2012)
(a) 32 (b) 64 (c) 84 (d) 96
€ Acube of edge length 6 cm. ; then its total area = ·--------- cm?
(a) 36 (b) 72 (c) 144 (d) 216
d If the perimeter of one face of a cube = 4 cm., then its total
area = -......... em? (Et-Dakahlia 2011)
(a) 3 (b) 4 (c) 5 (9) 6
е The area of base of a cube is 49 cm? , then its lateral area
(Cairo 2017)
(Alexandria 2014)
(a) 392 (b) 294 (c) 196 (d) 98
f Acube of total area 150 cm? , then the length of its edge is ст.
(Luxor 2015)
(а)5 (b) 6 (c) 15 (d) 10
9 If the total area of a cube is 24 cm? , then its volume = <... cm?
(El-Fayoum 2012)
(а)8 (b) 2 (c) 4 (d) 16
h A cube its lateral area = 36 cm? , then its volume = = cm?
(Souhag 2013)
(a) 27 (b) - 27 (c) - 1 (d) 2
i Acube: its volume is 1000 cm? , then its lateral area = ст?
(Damietta 2016)
(a) 600 (b) 500 (c) 400 (d) 200
j Acube-shaped box ; without a lid , has
(а) 4 (b) 5 (с)6
k А cube without a lid of edge length 3 cm. ; then its total area =
(a) 54 (b) 45 (c) 36 (a) 9
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى EE
Unit Three
! The area of one face of the cube = -------- its total area. (Kafr El-Sheikh 2011)
(а) } Ы $ 4 (i
m If the edge length of a cube equals the side length of an equilateral
triangle whose perimeter is 18 cm. , then the lateral area of this
cube m?
(a) 36 (b) 144 (c) 216 (d) 180
Ш A container water tank is in the form of
a cube whose inner length is 1.5 m.
It is wanted to paint it to prevent the rust.
The cost price of one square metre is L.E. 15
Calculate the cost of painting. (Giza 2017)
In the opposite figure :
A solid consists of two sticking cubes: the length of
the edge of one of them is 2 cm. and that of the other
is 1 cm. ;then find the total area of the solid.
BA factory has a quantity of carton paper of area 120 m? It is required
to make cube-shaped boxes of it › each box is of edge length 30 cm.
Find the number of boxes which can be obtained given that 10 96
of the area is lost.
Calculate the lateral area and total area for each of the following cuboids :
a m b ш
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى EE
PL] The perimeter of the base of a cuboid = 24 cm. and its height = 10 cm.
Calculate the lateral area. (Giza 2014)
ГЕ] A cuboid ; its length is 6 cm. , its width is 4 cm. and its height is 8 cm.
Find its lateral area and its total area. (Alexandria 2017)
19 Calculate the lateral area of a cuboid , its base is a square of side length
6 cm. and its height is 10 cm. (El-Kalyoubia 2015)
EI] ^ cuboid of a square base with side length 8 cm. and its height equals 10 cm.
Find :
а Its lateral area. b Its total area. (Et-Dakahlia 2012)
13 A cuboid-shaped box with a square base whose side length is 9 cm.
and its height is 20 cm. Calculate the lateral area and the total area.
(El-Fayoum 2012)
Т] A cuboid whose total area = 132 cm? and its lateral area = 112 cm?
Find the area of its base. (Giza 2016)
(a cuboid whose lateral area is 160 cm? and the dimensions of its base are
7 cm. and 3 cm. Find its height. (Ismailia 2014)
EI] A cuboid with a square base whose perimeter is 20 cm. and its height is 8 cm.
Find :
a The lateral area. b The length of its base side.
€ The total area. (Ismailia 2012)
EJ The perimeter of the base of a cuboid = 32 cm. and its height = 10 cm. ;
if the length of the base = 9 cm. Find the lateral area and the total area of
the cuboid. (El-Gharbia 2015)
£ A cube is of edge length 10 cm. and a cuboid whose length is 8 cm. »
its width is 5 cm. , and its height is 17 cm.
Calculate the difference between their lateral areas.
ат)
Tues] هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى
Unit Three
Ba tin, shaped as a cuboid without a lid , іѕ 18 cm. long and 7 cm.
wide and its height is 12 cm. Calculate its lateral area and its total area.
(Ismailia 2011)
LJ A box without a lid whose length is 16 cm., its width is 7 cm. and its
height is 19 cm. Calculate its lateral area and total area. (El-Kalyoubia 2016)
ЕА cuboid base is a square of side length 32 cm. and its height is $ the side
length of its base. Find its total area.
Е The volume of a cuboid is 180 cm.? and the dimensions of its base are
5 cm. and 1.2 dm. Find its total area.
When folding the opposite figure › complete :
a The formed solid is
b The lateral area of the solid =
€ The total area of the solid = =-=-
(Souhag 2013)
(ED complete the following table considering the unit length measured by cm. :
Cuboid Width Length Height | Lateral area | Total area
A 6 9.5 | 8
5 dias 4 120
C н 7 220
Complete :
а The lateral area of the cuboid = Б nt (Souhag 2017)
b The total area of the cuboid = ~--~ (Ismailia 2014)
€ Ifthe perimeter of the base of a cuboid = 20 cm. .
2
and its height = 9 cm. ; then its lateral area = -Cm (El-Dakahlia 2011)
d A cuboid of length 6 cm. , width 4 cm. and height 10 cm. د
then its lateral area = ст? (Cairo 2013)
)173
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتد اوله على مواقع أخرى EE
A cuboid has a square-shaped base with side length 20 cm.
and a height 25 cm. ; then its lateral area > =-= cm?
If the lateral area of a cuboid is 60 cm? and its base area is 8 cm? »
then its total area = =-= cm?
А cuboid of length 7 cm. , width З cm. and height 8 cm. د
then its total area = cm?
If the lateral area of a cuboid = 120 cm? and the dimensions for the
base are 4 cm. and 6 cm. ; then its height = --------- ст. (Alexandria 2012)
4 hoose the correct answer from the given ones :
a The lateral area of the cuboid = the perimeter of the base x
(Suez 2016)
(a) height (b) width (c) length (d) volume
b The lateral area of the cuboid with length is 3 cm. , width is 2 cm. and
height is 4 cm. = - 5 (El-Beheira 2013)
(a) 20 (b) 24 (c) 40 (d) 52
€ The lateral area of a cuboid with base in the shape of a square with side
length 8 cm. and the height of the cuboid is 5 ст. = + cm?
(a) 40 (b) 80 (c) 160 (d) 240
The total area of the cuboid with length is 12 cm. د width is 6 cm. and
height is 4 cm. = - cm?
(a) 216 (b) 36 (c) 360 (d) 288
The height of the cuboid whose lateral area is 120 cm? and the
dimensions of its base are 6 cm. and 4 cm. -Cm. (El-Gharbia 2014)
(a) 5 (b)6 (c) 12 (d) 2.5
If the total area of a cuboid = 32 cm? and its lateral area = 12 cm? ,
then the area of one of its bases = =-=- cm?
(a) 32 (b) 20 (c) 18 (d) 10
The dimensions of a base of a cuboid are 4 cm. and 3 cm. and its lateral
area = 140 cm? , then its volume = cm?
(a) 1680 (b) 120 (c) 168 (d) 60
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى Tues]
Unit Three
A swimming pool whose base is of dimensions 40 m.. 10 m. and its height
equals 2.5 m. Calculate :
2 Its lateral area. b Its total area. (El-Monofia 2011)
(O The cuboid-shaped box of a truck with inner
dimensions 4 , 2.5 and 1 metres, as shown in
the figure, is painted internally. If each square
metre of the box to be painted costs L.E. 8
Calculate the paint cost.
ДД] EJ A truck box is in the form of a cuboid, whose
inner dimensions are 5 m. , 2.5 m. and 1.6 m.
It is wanted to paint the inner box with paint د
the cost price of one square metre is L.E. 12
Calculate the cost of paint.
Ш A truck box for carrying goods is in the form of cuboid whose inner
dimensions are 4 m. , 2.5 m. and 1.8 m. It is wanted to cover its sides
and ceiling with a sheet iron, the cost price of square metre is L.E. 15
Calculate the cost of required sheet iron.
А cuboid-shaped box without а lid has a base with inner dimensions
1.6 m. long د 1.5 m. wide and 80 cm. high. We want to cover it from inside
with iron sheets that cost L.E. 10 per square metre.
Find the cost of the iron sheets needed.
EDA water tank is in the form of a cuboid of inner dimensions З m. ,2 т.
and dd m. The required is to paint it internally. If the cost price of each
square metre is L.E. 10 , then calculate the cost of painting of all the inner
@
0 هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى TEE]
surface of the tank.
: (4)
L3 A room whose length is 5 m. ; its width is 4 m. , and its height is 3.2 m.
It is wanted to paint its lateral walls and ceiling. The cost price of one
square metre to be painted is L.E. 8 Calculate the required cost knowing
that the room has 2 windows and a door whose areas are 8 m?
CJ A room of a squared floor whose length is 4 m. and its height is 3 m.
It has a door whose width is 90 cm. د and 2 m. high. It has two rectangular
equal windows of length 100 cm. and width 61 cm.
Calculate the cost of painting of the walls + given that the cost price of
painting one square metre is L.E. 9
Ga room has a square floor of side length 5 m. and height 2.8 m.
It has a door of width 90 cm. and height 2.20 m. and two windows each of
dimensions 1 m. and 60 cm. The walls and ceiling of the room are painted.
If the cost of one square metre to be painted is L.E. 10
Find the cost of painting the room.
(EJ The inner dimensions of a swimming pool
are 25 , 12 and 2.25 metres. It is necessary
to cover its inner floor and sides with
square-shaped ceramic tiles of side length
25 cm. How many tiles are needed ?
EJ rhe inner dimensions of a swimming pool
are 25 m. , 16 m. and 3.5 m. It is necessary
to cover its inner floor and sides with
square-shaped ceramic tiles of side length
20 cm. How many tiles are needed ?
LJ A swimming pool whose base is with dimensions 40 m. , 10 m. and its
height equals 205 cm. It is needed to be covered by ceramic with square
shape of side length = 25 cm. Find :
@ The number of boxes of ceramic is needed if each box contains
25 units of ceramic.
EE "هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتد اوله على مواقع أخرى
Unit Three
b The total cost if the price of one square metre of ceramic = 45 pounds
and the cost for covering 1 square metre by ceramic = 5 pounds.
L3 Youssef used a piece of cardboard in the form of a rectangle ; its length
is 1.2 m. and its width is 80 cm. to form a cube-shaped box whose edge
length is 30 cm. Calculate the remained paper area after forming the box.
If we double the dimensions of a cuboid , what is the ratio between the
new total area and the initial one ?
If the total area of the cuboid is 400 cm?. and the side length of its
square-shaped base is 10 cm., then find its height.
бе REEL
E The sum of the lengths of the edges of a cuboid is 136 cm. and the ratio
between the dimensions of its base is 3 : 5 , if its height is 1 dm.
Find its total area.
=) A cuboid in which : its length + its width = 16 cm., its width + its height = 14 cm.
and its height + its length = 18 cm. Find its lateral area and its volume.
The height of a cuboid-shaped water tank is 4 m.
` andthe perimeter of its base is 20 m. and the
ratio between its base dimensions is 3 : 7
Find the cost of painting the outer surface of
the tank د without its base with special paint
that costs L.E. 5 per square metre.
The total area of a cube-shaped piece of metal is 384 cm? It is melted and
shaped like a cuboid whose base dimensions are 16 cm. and 2 cm.
Find the total area of the cuboid.
A cube with edge length 12 cm., a part of it is Zi g
cut to form a cuboid whose side lengths are fd
3 ст.» 2 cm. and 1 cm., find the total area of
the remained part of the cube.
fm.
стат ә يسان ر ciii, gall (177)
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى [ЖЕЕ]
Test оп Unit Three
Answer the following questions :
Е Choose the correct answer :
а The total area of the cube = area of one face х ===
(4 or 6 or 8 or 12)
b The image of the point (3 , — 4) by translation (X — 1 ,у + 4) is ==-
((4 +8) or (2.0) or (2.8) or (3,0) )
€ IfA(6 , 1) and B (6 , 4) , then the length of the line segment
AB = units. (—4-or 4 or 5 or 3)
d The area of the circle whose radius length is 7 cm. = cm?
(Consider x = 22) (88 or 44 or 49 or 154 )
© The lateral area of the cuboid = the perimeter of the base x =.
(height or width or length or volume )
B Complete each of the following :
The area of the circle = ---------
A cuboid of length 6 cm. , width 4 cm. and height 10 cm. ; then its
lateral area = ---- em?
The total area of a cube of edge length 7 cm. = cm?
The image of the point (2 , 3) by translation (— 1 ; 1) is ~
In the opposite figure :
Ais the image of A by a translation
» its magnitude is «< ст. in the
direction of
22| هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى D
Unit Three
[|
A circle; its circumference is 88 cm. , calculate its surface area.
(Consider 7 = 3.14)
In the coordinates plane, draw the triangle ABC where A (0 › 1)
> B (2.3) and © (- 1.4) , then find the image of the triangle ABC by
translation (2.3)
A case in the shape of a cuboid its base is a square of side length 8 cm.
and the height of the case is 22 cm. Calculate its lateral area and its
total area.
In the coordinates plane ; draw the rectangle ABCD where A (4 , 2) »
B (4 ,4) , © (154) and D (1,2) » then :
(1) Draw its image by translation (X + 2 , y + 2)
(2) Calculate the perimeter of the image of the rectangle ABCD
The sum of edge lengths of a cube is 60 cm. Find :
(1) The lateral area of the cube.
(2) The total area of the cube.
(3) The volume of the cube.
In the opposite figure :
XYZL is a rectangle, its length is 10 cm. د
its width is 7 cm. Calculate the area of the
shaded part. (Consider л = 22)
179.
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى EE
O
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Technology of Unit Three
Activity. Finding the lateral area and the total area of the cuboid
by using Excel program
The dimensions of cuboid
Lateral area | Total area
The length The width The height
4cm. З cm. 5 cm.
10 cm. 8 cm. 6 cm.
6 cm. 6 cm. 12 cm.
12 cm. 4.5 cm. 5 ст.
| 9.6 cm. 7.5 cm. 6.3 cm.
Procedures of activity : |
[1] Click "Start" button from
the task bar.
[2] From the menu
"All programs" select
"Microsoft Office" . then
select "Microsoft Excel". |
|
4
[3] Write the dimensions of 2
each cuboid in the defined 12
cells in the excel sheet as
figure (1)
180
length width height Lateral area
3 5
8 6
в 12
45 5
T5 63
Fig. (1)
TEE العمل خاص بموقع ذاكرولى التعليمى ولا یسح بتداوله على مواقع أخرى m
Total area
а. Pe كناب اتمساصرا
Unit Three
[4] To calculate the lateral area
and the total area of cuboid
First : select O4 cell and
type = (R4 + Q4) #2* P4 ,
then press enter key.
Second : select N4 cell and
type = O4 + 2*R 4« Q4
»then press enter key.
[5] To calculate the lateral area
and the total area of the
other cuboids , select the
two cells O4 and N4 then
drag to apply the properties
of O4 and N4 on the other
cells , then the results will
be shown as in the figure (3)
The dimensions of a cuboid
length width height Lateral area Total area
4 3 5 70 94.
10 8 в
6 6
12 5
The dimensions of a cuboid
length width height Lateral area — Total area
(181)
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى EE
Activity of Unit Three
(1. C1 Bring a sheet of cardboard (Bristol) ,
then cut a square from each corner whose
side length is 15 cm. to form the opposite
figure. Fold the shape and use the glue to
form a cuboid without a lid. Use your
instruments to calculate its lateral area
and its total area.
The opposite figure represents an unfolded
cube. Copy the shape with the same given
dimensions on a lattice. Show how you
can fold it to get a cube : then calculate :
a Its total area.
b Its volume.
' هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى [ст
UNIT AIMS
Statistics and
probability
ó)
Lessons of the unit :
1. Representing the statistical data by using the circular sectors.
2. Random experiment.
3. Probability. } n
Test on unit four.
© Technology of unit four. 7
© Activity of unit four. 9
By the end of this unit, student should be able to
+ recognize the circular sector (minor - major). ^ ^ recognize the random experiment and the 1
+ know that the sum of measures of the sample space
accumulative angles at the centre of the circle + write the sample space of a random experiment. | إلى
is 360° © à
cognize the event in a random experiment.
represent the statistical data by using the = calculate the probability of occurrence of
circular sectors (a pie chart). an event. Ф
e
8
١
EE هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى ju
Woe oe (ERES
»
Representing the statistical data
by using the circular sectors
* You have studied before how to represent data by bar graph , broken line 9
9, graph or double bars graph. '
* Now; we are going to present another type of graph called "pie chart" in q 1
which a circle is divided into sectors that each represent a proportion of the X
А ў
\ 4 whole. It is a good way to show relative size of data. 4
{ —— x
* When different items of data are presented on a pie chart, you can easily
do a quick comparison between these items, and also between any item
| and the total.
v
For example : i
4 In the opposite figure :
Each of the coloured part and the uncoloured part of ву
the circle M represents a circular sector , where :
* The coloured part AMB is called "minor sector" Í
EL
D
p م
because its area is less than 4 the area of the circle. ` /
^ ت
The uncoloured part AMB is called "major sector" [^ *
У because its area is more than + the area of the circle. 4
| 5
>
184 6
ТЕШЕ أ هذا العمل خاس بموقع ذاكرولى التعليس ولا سمج بتداوله على مواقع أخرى ١
I LIC الصف الما كنات
»
Unit Four
(1) Each circular sector has an angle whose vertex ie the centre
of the circle which is called a “central angle”.
(2) The sum of the measures of angles accumulating around at
a point as the centre of the circle ie equal to 360°
(3) © A quarter ( 1 ) of the area of а circle represente 25 %
of the whole data.
* م half ( z ) of the area of a circle represents 50 %
of the whole data.
* Three quarters ( 2 ) of the area of a circle represent 75 %
of the whole data.
Example (1 J =
The opposite figure represents the different
activities which Sally does during a day.
Study the figure » then answer the following
questions :
[a] Find the percentage of the time that Sally spends at school.
[b] Find the percentage of the time that Sally spends in sleeping.
[c] Find the percentage of the time that Sally spends in other things.
Solution
[a] 20 % [b] 30%
[c] 100 % — (30 % + 20 96 + 10 % + 10 96) = 30 %
[d] Studying : playing (or : sleeping » other things)
(rein ۲ ترم / шшш \/ әш رياضيات wala! (185)
ر هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى Tus]
The opposite figure represents the percentages of
the favourite subjects of 200 pupils in a school.
Answer the following questions :
[a] What is the ratio of the pupils who prefer English ?
[b] What is the ratio of the pupils who prefer Science ?
[c] What is the ratio of the pupils who prefer Mathematics ?
[d] Which sector represents the greatest ratio ?
[e] Which sector represents the smallest ratio ?
[f] Find the measure of the central angle of Maths in degree.
[g] How many pupils prefer studying English ?
Solution
[a] 20% [b] 10%
[c] 100 96 — (15 % + 20 96 + 25 % + 10 96) = 30 %
[d] Maths [e] Science
[f] The measure of the central angle of maths = 30 96 of 360°
= Ж x 360° = 108°
[g] The number of pupils = 20 % of 200 = E x 200 = 40 pupils. | —————
The opposite figure shows the percentages of time
that Enas spends in studying different subjects.
Complete : 20%
[a] The ratio of the time that Enas spends in
studying maths is
[b] The measure of the central angle of science in degree is ---
[c] The subject that needs more time is
T
EE هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى
Unit Four
Example(3 — —
The following table shows the percentages of the number of hours
that Marwa studied in different subjects in a week :
Subject Arabic Maths Science English
Percentage 10% 40% 20% 30 %
Represent these data by a pie chart.
Solution
First :Find the measure of the central angle which represents
the percentage of each sector as the following :
* The measure of the central angle for Arabic — 19, х 360° = 36°
* The measure of the central angle for Maths = p x 360° = 144°
* The measure of the central angle for Science = 20; x 360° = 72°
* The measure of the central angle for English = ES x 360° = 108°
Draw a circle of a suitable radius › with centre М
: Draw the radius MA , use your protractor to draw the central angle
AMB with the measure of 36° The sector AMB represents Arabic.
Similarly . draw / BMC of measure 144°
The sector BMC represents Maths ,
using the same method , draw the other
two sectors ; then you will have the
opposite figure.
187)
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى EE
Example (4
The following table shows the favourite TV programs for some pupils :
TV program Sports News Series Movies
Number of pupils 15 5 10 30
Represent this data by a pie chart.
Solution |
The sum of pupils = 15 + 5 + 10 + 30 = 60 , then:
* The measure of the central angle for sports
= 15 ° = 90°
= 60 * 360° = 90
* The measure of the central angle for news = a x 360° = 30°
* The measure of the central angle for series = 19 х 360° = 60°
• The measure of the central angle for movies = 30. „360° = 180°
0
60
The following table shows the percentage of the production of egg
in three farms monthly :
Farm | Fist | Second | Third
Percentage of egg production | 50 % 20% 30 %
Represent these data by a pie chart.
)88(
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى EE
A TTL | ENEE)
Exercise 1H
Representing the statistical data
by using the circular sectors
LJ From the school book
п The opposite figure shows the percentages of sales of different types
of books. Complete :
а The sales percentage of science books is =-
b The least sales percentage is in ıı...
€ The ascending order of books types according to
the percentage of sales
The opposite figure shows the percentages of family spend in different
purposes. Study the figure » then answer the following questions :
a Whatis the ratio (in fraction) of clothing to house rent ?
b What is the ratio (in fraction) of clothing to food ?
€ Whatis the ratio (in fraction) of clothing to savings ?
d Find the measure of the central angle of clothing
in degrees.
B {Ш The opposite figure shows the favourite hobbies for the pupils of one
of the classes in the sixth primary » study the figure » then answer :
What is the ratio of the theatre with respect
to all hobbies ?
What is the ratio of the broadcast with respect
to all hobbies ?
What is the ratio of the rangers with respect to
all hobbies ?
d What is the measure of the central angle of the sector of the music ?
e What is the hobbies that the least pupils prefer ?
What is the hobbies that the most pupils prefer ? ,
189)
هذا العمل خاص بموقع ذاكرولى التعليمى ولا
a {1 The opposite figure shows the distribution of the natural components of
the earth's surface » study the figure » then complete the following table :
The components of | Water natural 5 е
5 Vallies Mountains
the earth’s surface supplies
The percentage of
the forming
What is the component which represents
the smallest ratio of the earth's surface ?
What is the component which represents
the greatest ratio of the earth's surface ?
What is the measure of the central angle Hills E] Water E]
of the sector of the vallies ? Mountains [E]. Vallies 5
at the zoo. Study the figure • then answer the questions :
a Which animal site is favoured by almost
half of the people ?
b Which two animals are favoured by almost
the same number of people ?
с What is the percentage of lion site ?
d Whatis the percentage of donkey and snake sites ?
[6] Forty students were surveyed about their favourite fruit. The opposite
figure represents the outcome of the survey. Study the figure » then
answer the question :
a How many students like apples ?
b How many students like bananas and peaches ?
с Whatis the ratio (in fractions) of students who like
apples and oranges ?
d Arrange the percentage of students in a descending
order according to their favourite fruit.
(190)
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى EE
Unit Four
The following table shows the percentage of the production of meat in
( 3 slaughter houses during a month :
Slaughter house 45 gnd 3rd
The percentage 20% 30% 50%
Represent these data by circular sectors. (Red Sea 2016)
a 1 The following table shows the percentage of the egg production in
three farms , a merchant collected these eggs to distribute them on the
grocery stores , represent these data by using the circular sectors :
The farm First | Second | Third
The percentage of the production | 25% 35% 40%
(Alex. 2014)
LA The following table shows the ratio for producing electronic sets :
Set kind n and зч | سه |
The ratio of production | 30% | 15% | 40% | 15% |
Represent these data by a pie chart.
5 Ш The following table shows the percentage of the production of one
factory of 4 kinds of electric sets :
TV Washing
Type of the set machine
Refrigerator | Cooker
Amount of the production 35% 20% 15% 30%
Represent these data by using the circular sectors. (Damietta 2011)
The following table shows the percentage of the students participated in
the school activities :
Activity Cultural Sport Social Art
Percentage of students 5% 45% 15% 35%
Represent these data by a pie chart. (El-Menia 2017)
(491)
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى Tues]
( The following table shows the percentage of the favourite subjects for
the sixth primary in one of the schools through their questionnair :
Subject : Arabic Maths | Science | Social studies
The percentage of
% 2 25 % 159
the number of the pupils 9979 5% 5 S
Represent these data by circular sectors. (Ismailia 2014)
The following table shows the percentage of the time Ayman spent
studying some subjects :
Subject Arabic Maths Science | Social studies
Percentage 25% 35% 15% 10%
Represent these data by a pie chart.
ij The following table shows the percentage of the favourite sports of
students in your class :
| The favourite sport | Football | Basketball | Volleyball | Swimming | Ping-Pong
| Percentage | 45% | 9% | 24% 10% | 12%
Represent the previous data by using the circular sectors.
The following table shows the product of 4 farms in a month :
The farm | 1” gral ач
Percentage | 40% 25% 20%
a Represent these data by using the circular sectors.
b If the total product of these farms in a month was 1200 chicken.
Find the product of the 1° farm in this month. (Qena 2013)
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى Tues]
Unit Four
The following table shows the rate of the score of 200 students in one
school of Cairo governorate :
Excellent Good Pass
25 96 10%
Rate Weak
15% 50 %
Percentage
а Represent these data by a pie chart.
b Find the number of excellent students.
The following table shows the percentage of chicken production for three
farms during October :
First Second Third
50 %
The farm |
The percentage of production | 25%
a Complete the previous table.
b Represent these data by the circular sectors.
(El-Fayoum 2013)
The monthly income of a family is 480 pounds , the table shows the
percentage of its expenditure :
Expenditure Rent Food Others | Saving
Percentage 20 % 60% | 5* |:
a Complete the table.
b Represent these data by a pie chart.
8 The following table shows the percentages of the time Waleed spent
studying some subjects during a week :
Arabic Maths Science
40% 15%
Subject Social studies
Percentage 25%
a Complete the table.
b Represent these data by a pie chart.
(тез تيرم ؟ шы! \/ رياضيات لغات wall (493
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى EE
Б K1 The following table shows the ratio for producing chickens in four farms
in a month :
Farm 4st 2nd ard 4th
The ratio of production
a Complete the table.
b Represent these data by a pie chart. (El-Kalyoubia 2011)
(E) The following table shows the ratio for producing chicken in four farms :
1e E 38 4th
The farm
The ratio 30 % 40%
a Complete the table.
b Represent these data by the circular sectors.
€ Which farm produces more chicken ? (Kafr Et-Sheikh 2011)
Anania is a clerk in an institution. She contributes with her husband by her
salary as follows :
25 % for house rent , 50 % for food and expenses and 25 % for savings.
Represent those data by using the circular sectors. (The New Valley 2011)
The monthly income of a family is L.E. 1200 . the family spends 25 % of its
income on rent , 50 % on food . 15 % on others and saves the rest :
a Represent these data using a pie chart.
b Find the capital which this family saves monthly.
ea KJ One of the families spends its salary as the following : 40 % for food
» 20 % for house rent , 30 % for expenses and saves the remainder د
represent these data by using the circular sectors . then answer the
following :
* If the family monthly income is L.E. 900, so how much does the family
save in the year ?
» Another family spends its monthly salary by the same way and saves
L.E. 70 monthly . so what is the monthly salary of that family ?
(494)
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى Tus]
Unit Four
The following table shows the number of studying hours that Mohamed
has done in a week :
Subject Arabic | Maths | Science | English | Social studies | Total
Number of hours 9 10 6 ГА 4 36
Represent these data by a pie chart. (Souhag 2013)
LJ The following table shows the favourite TV programmes which
the pupils of one of the classes in the primary six watch as the following :
Number of hours 9 5 4 7 11
Kind of programme | Entertaining | Cultural News Drama Sport
Represent the data by using the circular sectors, then answer the following
questions :
What is the programme that the most of pupils prefer. also the least of
pupils prefer ?
(The following table shows the Suez Canal income in million Egyptian
pounds during some years :
Year 2001 2002 2003 2004
The income in million 120 180 175 525
Represent these data by a pie chart.
The following table shows the percentages of success in a Maths test in
five different classes of grade six :
Grade six classes 6A 6B 6c 6D 6E
Percentage of success | 95% 90 % 75% 80% 65%
Represent the percentage of success of each class by a separate pie chart
» then answer the following questions :
a Which class has the greatest percentage of success ?
b Which class has the lowest percentage of success ?
с What do you suggest to raise the percentage of success of the class
having the lowest percentage ?
Go
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى EE
Complete each of the following :
a The sum of the measures of the accumulative angles around the centre
of the circle = =- ? (Et-Katyoubia 2014)
The measure of the angle of the circular sector whose area
represents i from the area of the circle = === 2 (Beni Suef 2013)
The measure of the angle of the circular sector whose area
represents + from the area of the circle = ---------- و (Qena 2017)
The measure of the angle of the circular sector whose area
represents 4 from the area of the circle = E (Ismailia 2016)
The measure of the angle of the circular sector whose area
represents 4 from the area of the circle = :........ е (Assiut 2012)
In the opposite figure :
The measure of the central angle of
the shaded circular sector equals
(Cairo 2016)
Ш The opposite figure represents the percentages
of distribution of the sports activities for the pupils
of a school , their number is 960 pupils.
(1) The percentage of the pupils
Basketball
participated in handball = <... % vie
(2) The number of pupils who participated in football activity
يع pupils.
(3) The measure of the central angle of the sector representing
the pupils who are participating in volleyball activity
(Aswan 2015)
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى EE
M ILC eem]
box contains 3 balls ! Í Abox contains م
(red, green, black) 3 3 red balls
l'm not sure Ay p ^y
if the selected ball Y: begh 2 УХ ‹
چ N that the selected ball \
| UR 1 will be red. \ |
1 or black i
Ф This is 4 ы: This is NOT 1
a random experiment a random experiment
| Definitions : |
* The random experiment :
It ig an experiment in which we can determine all ite possible outcomes before
carrying it out » but we can't predict in certainty which of these outcomes will
occur when the experiment ie carried out.
| * Sample space (outcomes space) :
It ig the set of all possible outcomes for a random experiment.
lt ig ugually denoted by the symbol (S) and the number of all elements of
the sample space ig denoted by n (S)
197
E على مواقع أخرى Ан يسع MY eda sif العمل خاص بموقع m
| كتاب اتمعاصر | [той الصف السادس الايد اتآ |0"
Examples of random experiments :
Tossing a coin is a random experiment, because before
tossing the coin you do not know the result.
Rolling a die is a random experiment; because before
rolling the die you do not know the result.
Drawing a marble from a bag containing marbles different
in colour (or size) is also a random experiment. because
before drawing the marble you do not know the result.
Example (1)
Write the sample space of each of the following random experiments
and give the number of its elements :
[a] Tossing a coin once and observing its apparent face.
[b] Rolling a die once and observing the number appearing
on the upper face.
[c] Drawing one card from five cards numbered from 10 to 14 and
observing the written number on the card.
[d] Drawing a ball from a bag containing : one black ball , one red ball »
one yellow ball and one white ball and observing the colour of
the drawn ball.
[e] Playing a football match between the team of your school and the team
of another school , and observing the possible results of the team of
your school.
[f] Choosing a prime number less than 18
Solution |
[a] The possible outcomes are : head (H) or tail (T)
58= {н.т}, п (5) =2
[b] The possible outcomes are: 1,2 ,3 ,4 ,5 6
25={1,2,3,4,5,6},п(5)=6
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى Toss]
Unit Four
[c] 5 = (10. 11.12.13 ,14( مء (S) =5
[d] S = {black , red , yellow , white} >n (5) = 4
[e] S = {win , loss , draw] مه (S) = 3
[f] S2 (2.3.5.7 11.13. 17] »n(S)=7
Example (2) —— —
Write the sample space of tossing two distinct coins once and give
the number of its elements.
Solution IS соп 2" coin bei]
We can get the elements of the sample кы нн
space by using the tree diagram as
shown in the opposite diagram.
.. S= (HH; HT TH , TT} 5n(S)=4
Where :
HH means that the result of tossing the coins in
the first coin is head and the second is head,
HT means that the result of tossing the coins in
the first coin is head and the second is tail and so on.
o The sample space of tossing two distinct coins (different in colour or
size or shape ; ...) simultaneously is the same as the sample space of
tossing one coin twice one after the other.
© The sample space of rolling two distinct dice is the same as the sample
space of rolling a die two consecutive times.
E
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى EE
(2)
Write the sample space of each of the following random
experiments and give the number of its elements :
[a] Drawing a ball from a bag containing : one white ball , one green ball |
and one red ball and observing the colour of the drawn ball.
[b] Tossing a coin twice.
dl
| Example (3
A box contains three balls. One of them is white , the second is red and
the third is black.
The experiment is drawing two balls one after the other with replacement
and observing their colours. (st 2m Sample.
State the sample space and give the drawing drawing space (S)
number of its elements.
|
Solution
Let W denote the white ball , B denote the
black one and R denote the red one.
Using the opposite tree diagram: we get :
S = {WW , WR; WB, BW; BR ;
ВВ , RW , КК , RB} ,п (5) = 9
BRE DHE OH
| Example (4
From the set of digits (4 , 5) › a number is formed from two digits.
Determine the sample space of this experiment
Tens Units Sample
and give the number of its elements.
digit digit space (S)
Solution | * ы
Ee @ x
From the opposite tree diagram :
S = {44,45,54 ,55} >n (S)=4
5
84| 4
55 5
=
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتد اوله على مواقع أخرى EE
_ wwwizakrooly'com | | Maths | (MESSI
Exercise 15
Random experiment
LJ From the school book
п Determine the sample space of each of the following random experiments ›
and give the number of its elements :
a Choosing a card from 5 cards numbered from 3 to 7 and
observing the written number on the card.
Choosing one of the digits of the number 23791
Choosing an even number included between 21 and 29
Choosing a prime number included between 10 and 20
Drawing a ball from a bag containing : one green ball , one yellow ball
and one black ball and observing the colour of the drawn ball.
LH A box contains 9 identical cards having the same
colour and numbered from 1 to 9 9 4 Pun
8
6
Write the sample space for this experiment. 5 2
E^ bag contains 4 identical cards having the same colour and numbered
from 30 to 33 Write the sample space for this experiment.
u LA If the random experiment of visiting one of your relatives to know
the gender of his newly-born child.
Write the sample space of this experiment.
E Write the sample space of tossing a coin twice in succession and
observing the sequence of heads and tails showing the number of its
elements.
[e] {п the experiment of selecting a ball from a box containing 3 red balls,
4 yellow balls all of them are equal in volume: observing the colour of the
selected ball , write the sample space of this experiment.
(en تيرم ؟ / uas ١/ йш رياضيات аја (201)
ر هذا العمل خاص بموقع ذاكرولى التعليمى ولا یسح بتداوله على مراع أخرى тз
١ 1
In an experiment of getting a 2-digit number using the digits 1 and 2
/ Write the sample space of this experiment.
a From the set of digits [3 ,4 ,9} .a number is formed from two digits.
Determine the sample space of this experiment and give the number
of its elements.
О л family has three children (there are no twins among them).
Write down the sample space (S) of the gender (boy or girl)
of each of them ordering them according to their ages.
Determine the sample space of tossing three distinct coins once and
observing the sequence of appearance of heads and tails.
[11] A die is designed such that two faces have the number 1 , two faces have
"the number 2 and two faces have the number 3 , this die is thrown once
and the apparent number on the upper face is observed.
Write down the sample space and give the number of its elements.
Ba coin is tossed , then a die is thrown and the upper faces of the coin and
the die are observed.
Write down the sample space.
Complete each of the following :
a LƏ The random experiment is ·------:--
b «is the set of all possible outcomes for a random experiment.
(Suez 2015)
In the experiment of tossing a regular coin and observing the appearing
face , set of sample space S = =-=- , n (S) = ==- (Giza 2012)
d The sample space for tossing a coin twice = =-
е The sample space for rolling a dice once = ·------·
(202)
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى EE
Probability
П
| The event : In a random experiment, an event is а subset of the sample space. y
For example :
In the random experiment of rolling a die once and observing the apparent
x number on the upper face , we have :
NI 8-(,2,3,4,5,6)
D Any subset of S can be considered as an event as :
* {5} is the event of getting 5 on the upper face of the die.
© * (2,4.,6) is the event of getting an even number on the upper face of the die.
* (I, 3 , 5) is the event of getting an odd number on the upper face of the die.
Oy Probability of occurrence of an event
In a random experiment with a sample space S if each element of S
ё ( hae the same chance to occur, and if A C S , then the probability that |
the event A will occur is :
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتد اوله على مواقع أخرى
الصف السادس الابتدانى | (ARR) | كتاب alas
ple (1
If a fair die is thrown once and we observe the number on the upper
face , find the probabilities of each of the following events :
[a] A is the event of appearance of a number greater than 4
[b] B is the event of appearance of an even number. ав а
[c] С is the event of appearance of the number 5
[d] D is the event of appearance of the number 7
[e] E is the event of appearance of a number less than 7 =
Solution
58={1,2,3,4,5,6},п(8)=6
[ајА = {5,6} ,п(А)=2 А P(A) =
[b] = {2,4,6} ,п (В) * 3 ТР (В) =
oj- ماه olv
"
[c] C= {5} » n (O) - 1 ЕР (QE
[d]D={ }or ® ,n (0) = zero
- P(D)= 2 = zero (The impossible event)
[е]Е={1,2,3,4,5,6},п(Е)=6
(E) = $ = 1 (The certain event) من
E атта
A fair die is thrown once. Find the probability of each of the
following events :
[a] А: The event of getting the number 6
[b] B : The event of getting a number less than 3
[c] С: The event of getting an odd number.
[d] D : The event of getting the number 8
[e] E : The event of getting a prime even number.
[f] F : The event of getting a number greater than 1
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى [сте
Unit Four
@ The impossible event is the event which cannot occur.
• The impossible event = Ø while the probability of the impossible
event = zero
.. P(g)-0
©) The certain event (sure event) is the event whose outcomes are all
the possible outcomes.
٠ The certain event = S while the probability of the certain event = 1
4s P(S)=1
e The possible event : some of outcomes of the experiment.
i.e. The probability of the possible event proper fraction.
So; the probability of any event is not less than zero and it is not more than 1
i.e. For any event A , we found that : 0 > P (A) > 1
=
A jar contains 9 similar balls “4 of them
are white » 3 are red and 2 are black"
A ball is drawn randomly.
Calculate the propability of each of
the following events :
[а] А: The drawn ball is white.
[b] B : The drawn ball is red.
[c] С: The drawn ball is green.
[d] D : The drawn ball is white or black.
[e] E : The drawn ball is not black.
(205)
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى EE
Solution
The number of white balls _ 4
The number of all balls ^ 9
[a] P (A) =
The number of red balls _ з
[b] P (В) = тһе number of all balls ^ 9
The number of green balls _ o
The number of all balls 9
[c] P (С) = 0
[d] P (D) = The number of white balls and black balls 4*2 6 -
= The number of all balls | 9 9
The number of balls which are not black _ 9-2 7
[e] (Е) = The number of all balls ^ 9 9
Example (3)
From the set of digits {3 .4 . 5] , form all possible two-digit numbers .
then find the probability of each of the following events :
[a] A "the event that the units digit is odd"
[b] B “the event that the tens digit is even"
[c] C “the event that the two digits are odd numbers"
[d] D "the event that the sum of the two digits = 8"
[e] E “the event that the product of the two digits = 20"
Solution
5 = {33 ,43 , 53 , 34 , 44,54,35 , 45 , 55} .n(S)- 9
[a] A= {33 ,43 ,53 ,35 ,45,55},п(А)=6 .. P(A)=
[b] = {43,44,45} >n (B) =3 .. P درق
[c] C = {33 ,53 ,35 , 55) >n (C)=4 2 P(C)=
[d] D = {53 ,44 ,35} ,n (D) =3 2 P(D)=
[e] E = (54 , 45} ,n (E) -2 ^ P(E) =
206
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى EE
€|w «|o oja ow وات
Unit Four
@ The probability can be written as a fractional form , decimal form or in
the form of percentage.
i.e. if the probability of an event = H „it can be written as 0.4 or 40%
© The sum of probabilities of all outcomes of a random experiment = 1
e If the probability of occurrence of an event A is P (A)
» then the probability that it doesn't occur = 1 — P (A)
For example :
If the probability of success of a student is 35 >
" 5 5 TAE
then the probability of his failure = 1 — 16 = jo
Example (4 2 E
A card is selected randomly from 30 cards numbered from 1 to 30
Find the probability of each of the following events :
[a] A : The selected card carries a number divisible by 5
[b] B : The selected card carries a number divisible by 9
[c] С: The selected card carries a number satisfying the equation : 2 X + 3 = 15
[d] D : The selected card carries a number satisfying the inequality : X — 5 2 22
[e] E : The selected card carries a number satisfying the inequality : 6 « X s 12
Solution
S=4{1,2,3,4,- 30} .. n (S) = 30
[a] A= {5 , 10, 15 ,20 ,25 ,30} .n(A)=6 PAHS
= = 5 = 3
[b]B={9,18,27}.n(B)=3 Р (B) = 35
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى EE
v2Xt*3215 є&2х=12
мХ=6 .. The S.S. = {6}
С={6},п(С)=1 = P (C) = зу
v X-5222 x227
г. The S.S. = {27 , 28 ,29 , 30}
.. D = {27 , 28 , 295 30}>n (D)=4
2Р0) = 35 =%
s6«xs12
г. The S.S. = (7 8,9, 10,11, 12)
nE-(7:859.10,11,12) م (E) -6
АРЕ) =
6 =1
30 5
Example (5 )
A bag contains 45 similar marbles. Wael drew one of them randomly
and found it green. If the probability of drawing a green marble = З.
Find the number of green marbles in the bag.
Solution
The number of all marbles = 45
Let A be the event of drawing a green marble , then P (A) = 3
. The number of green marbles _ з
The number of all marbles = 5
. The number of green marbles _ 3
45 5
2. The number of green marbles = 3 x 45 = 27 marbles.
E
FEE] هذا العمل خاص بموقع ذاكرولى التعليمى ولا بسمح بتداوله على مواقع أخرى
Unit Four
Example (6
Two players play in a football team.
During training » one of them kicked
20 penalty kicks and he scored 14 goals ›
and the other kicked 25 penalty kicks
and he scored 18 goals.
Which of them should you choose to
kick a penalty kick in the game? why?
Solution
* The probability that the first player scores а goal = i$ =0.
The probability that the second player scores a goal = i =0.72
50.72 > 0.7
.. The second player should kick the penalty because
his probability is greater.
(Y SD Y تيم gal V әш رابات wall] (209)
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى EE
mance) | Maths | (ESI
Exercise 16
Probability
L From the school book
п While throwing a fair die and observing the upper face , complete the following :
The probability of appearance of a number greater than 2 = (El-Gharbia 2017)
The probability of appearance of a number less than 3 = (Alexandria 2012)
The probability of appearance of an odd number = ===: (ei-Gharbia 2015)
The probability of appearance of the number 5 = = (Beni Suef 2016)
The probability of appearance of the number 6 = .......... (Aswan 2013)
The probability of appearance of the number 7 = =-
The probability of appearance of a number less than or equal to 6 = =-=-
The probability of appearance of a prime number = -........
The probability of appearance of a prime even number =.
The probability of appearance of a number divisible by 5 =
The probability of appearance of the number 5 or the number 6 = +--+
B A fair die is rolled once and the number of dots on the upper face is
observed , write down the sample space ; then find the probability
of the following events :
a Getting a number greater than 6
b Getting a number satisfying the inequality: 1 < X < 6
с Getting a number satisfying the inequality : 2 « X « 4
LA In the experiment of tossing a regular die once and observing
the number of dots on the upper face ; find the probability of :
a The event A , where A is the event of appearance of
a number less than 5
b The event B , where B is the event of appearance of
a number satisfying the inequality : X 2 3
210
"هذا العمل خاص بموقع ذاكرولى التعليمى ولا
EBES
Unit Four
LLI A box contains 8 white balls and 12 red balls all of them are symmetric د
( a ball is selected without looking inside the box ; find the following probabilities :
a The selected ball is white. b The selected ball is red.
c The selected ball is blue. (South Sinai 2014)
A box contains 4 white balls , 3 blue balls and 8 red balls , all of them
are symmetric د a ball is selected without looking inside the box ; find the
probability that the selected ball is :
a green. b notred. (Damietta 2013)
Ga box contains 8 white balls د 5 red balls and 7 blue balls , all balls
identical د if a ball is chosen randomly. Find the probability of :
a The chosen ball is white. b The chosen ball is not red.
с The chosen ball is white or red or blue. (Cairo 2013)
a A bag contains 25 balls (4 balls are yellow , 7 balls are red and the
remainder is black). If a ball is drawn randomly د find the probability that
the drawn ball is :
a black. b yellow or black.
c not yellow. d green.
e neither black nor yellow.
B CJ In the experiment of selecting a card at random from 7 equal cards
numbered from 1 to 7 , write the sample space , then find the probability of :
а The event A , where A is the appearance of a number less than 4
b The event В ; where B is the appearance of an odd number.
c The event C , where C is the appearance of a number more than 5
(Matrouh 2015)
(211)
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى EE
©
(E A box contains 10 identical balls numbered from 1 to 10 , one ball is drawn
at random. Write the sample space . then find the probability that the
drawn ball has :
а an even number.
b a number divisible by 5
с a prime number. (Cairo 2014)
Ш A basket contains 15 balls numbered from 1 to 15 , if one of the balls is
chosen randomly ; write the sample space for this experiment , then find
the probability that the chosen ball :
a carried an odd number.
b carried a prime number.
€ carried a number divisible by 3 (El-Beheira 2017)
C3 A box contains 10 cards numbered by the even numbers from (2 to 20).
one of the cards is selected at random. Calculate the probability of :
a The event А: appearance of a multiple of the number 4
b The event B : appearance of an even number.
с The event С: appearance of a number that is divisible by 3
(B^ number is chosen randomly from the numbers 1 ,2 535
Find the probability of the following events :
a A:the chosen number is a multiple of 3
: the chosen number is divisible by 7
: the chosen number is greater than 16 and less than 25
: the chosen number is a prime number less than 16
: the chosen number is divisible by 2 and 3
: the chosen number has 7 as a units digit.
©
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى EE
Unit Four
C3 By using cardboard ; cut 10 squared or rectangular equal cards
and have the same colour , then write a number in each one of them from
the numbers (1 to 10) , then put them in a bag that is not transparent and
mix them carefully ; choose one of them at random.
Calculate the probability of the following events :
a The event A : appearance of a number more than 7
b The event C : appearance of an odd number.
c The event В: appearance of a number that satisfies the inequality :
xs10
d The event D : appearance of a number that satisfies the equation :
x-4=2
C3 If the experiment of a student is chosen at random from a class of
40 students د 32 students succeeded in Maths test , 35 students
succeeded in Arabic test , find the probability of :
а The event A , where A is the event that he succeeded in Arabic.
b The event C ; where C is the event that he failed in Maths.
In the ideal student competition of one of the schools 63 students applied
for the competition . if the probability that one of the girls is an ideal student
is $ Find the number of girls who participate in the competition. (souhag 2013)
ЦЈ A box contains 80 similar balls. Some of them are red and the rest is blue.
If the probability of drawing a red ball is i > find the number of blue balls.
Ш ما a meeting for discussing the problems of the workers in a factory د
100 workers were attending from men and women. If the probability of
a man standing to show the problems of the workers is =
5
Calculate the number of the men and women in this meeting.
| Pe
mplete the following :
The event is a subset of the ·--------- (Souhag 2013)
In an experiment of throwing a fair die once , the event of getting
a number less than 2 is { (Giza 2014)
The probability of the impossible event = ~--~
and the probability of the certain event = =- (Beni Suef 2012)
d For every event A „ we find that <P (A) =
If S is the sample space of a random experiment ; then P (S) = =
(Matrouh 2017)
If n (S) = 12 , n (A) = 4 where AC S; then P (A) = (Ismailia 2012)
If a fair coin is tossed once ; then the probability of appearance of
a tail = 0...
In an experiment of forming a number from the two digits (2 , 3} »
the probability of getting an even number = ==- (El-Monofia 2017)
If the probability of occurrence of the event Ais 5 + then the probability
of non-occurrence of it = ---------- (Et-Fayoum 2017)
If the probability that a pupil solve a problem is 0.7 د then the number of
problems expected to be solved from the same kind from 20 problems
= (El-Menia 2016)
KE) Choose the correct answer from those given :
b
c
d
14)
If Ais an event of the sample space of random experiment
„then : 0 s P (A) < (Giza 2012)
(а) Ø (b)- 1 (c) 2 (d) 1
If P (A) = 1 » then Ais called event. (Ismailia 2012)
(a) a random (Ы) ап impossible (с) a possible (d) a certain
£ Ø is the empty set , then Р(@) = ~- (Port Said 2015)
(a)0 (b) 2 (c) 1 (d) 0.5
IfA=S , then P (А) = (Kafr El-Sheikh 2016)
(a) zero (b) 1 (c)2 (d)3
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى EE
Unit Four
Which of the following can be a probability of an event ? (El-Beheira 2017)
(a) zero % (b) 1.2 (c) 45 (d) 101 96
A coin was tossed once ; then the probability of getting a head
IS + x (Red Sea 2014)
(a) zero (b) 2 (0i (d) 0.25
A fair die is thrown once ; then the probability of appearing on the
upper face the number 3 = ---------- (Suez 2016)
(a) 0 (b) + (c) 2 (d) 1
If a fair die is rolled once; then the probability of getting
a number > 3 = eee- (South Sinai 2012)
(a) 1 (52 (e) 1 (9 1
LJ In the experiment of rolling a fair die once ; if A is the event of
getting a number less than 4 , then P (A) = =-=- (Qena 2013)
(a) 5. (b) 2 (oi (d$
LJ If a fair die is rolled once , then the probability of getting
a number > 6 = =- (Et-Katyoubia 2017)
(а) 2 (b) zero (i (9) i
When tossing a fair dice once ; then the probability of getting a number
divisible by 2 equals .......... (Red Sea 2017)
(a) zero (b) $ (c) 1 (d) 0.5
A bag has 5 red balls and 3 white balls. If the balls are similar and
a person draws a ball randomly , then the probability that the drawn
ball is white = -------..-
(a) 2 (b) 3 (e) 3 (a) $
A basket contains cards numbered from 1 to 20 ; if a card is drawn
randomly , what is the probability that the number written on it is
divisible by 6 ?
(а) 53; (Ы 3 (с) & (9) 5
If the probability of success of a student is 70 % , then the probability
of his failure = ----------
(a) 0.7 (b) 0.07 (c) 0.3 (d) 0.03
(215)
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى EE
©
о Ш A regular coin is tossed 1000 times د then the most expected
number to get a head equals (Alexandria 2013)
(a) 496 (b) 503 (c) 600 (d) 999
£3 In the experiment of forming a 2-digit number from the set of digits {5 » 6}
What is the probability :
a The event A : the units digit is an odd number.
b The event B : the sum of the two digits is 11
c The event C : the two digits are equal.
In one of the factories that produce saving energy electric bulbs ; if the
average of the daily production is 600 bulbs and the ratio of the damaged
bulbs is about 5 % of the production.
Complete :
а The number of the damaged bulbs during the daily production = =-
b The number of the working bulbs during the daily production = ===
с The probability that the bulb is working = =:
d The probability that the bulb is damaged = ----------
e If the production of the factory during few days is 2500 bulbs, so the
number of the damaged bulbs
(EB The set (2 ,3 , 5} is used to write a 2-digit number.
Find the probability of the following events :
a The tens digit is odd. b The units digit is odd.
c The sum of the two digits is 7 d The product of the two digits is 15
[23] Ш In the experiment of forming a number consisting of two digits without
( repeating the number using the set of numbers {1,2,3} Find :
a The probability of getting an odd prime number.
b The probability of getting an even number.
®
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى EE
Unit Four
A cube is designed such that each two opposite faces carry one of the
digits 1 , 2 and 3 , the cube is rolled and the apparent face is observed :
a Write the sample space.
b What is the probability that the number we get on
the upper face is 2 ? ШР
c Whatis the probability that the number we get on 5
the upper face is odd ?
LJ A class of 40 students has got a maths exam whose maximum mark
is 50 , if 30 students got less than 40 marks , and 10 students got
(40 up to 50). Calculate the probability of :
а The event А: where A is a student who has got less than 40 marks.
b The event B : where B is a student who has got a mark satisfying
the inequality : B 2 40
LJ In one of the fitness centres ; 10 ladies suffering from over weight were
waiting to meet the specialized doctor , if the weights of 4 of them are
between 100 , 110 kg. , the weights of the others are between 110 . 120 kg.
Find the following probabilities :
a Entrance of a lady of weight less than 110 kg.
b Entrance of a lady of weight more than 110 kg.
с Entrance of a lady of weight 90 kg.
In a class of 42 students , we found that 20 students play football ,
8 students play basketball and the rest of students play other sports.
One of those students is chosen randomly ; find :
a The probability that the student is one of those players who play
football.
b The number of students who play other sports if the total number of
students at school is 600
pals! (217) رياضيات لغات ایتداتی / تيرم Y (۳: ۲۸)
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى EE
Two players play in a football team. During training » one of them
kicked 21 penalty kicks and he scored 18 goals and the other kicked
32 penalty kicks and he scored 25 goals. Which of them should you
choose to kick a penalty kick in the game ? Why ?
[29] The opposite figure represents a spinner game :
a Find the probability that the pointer stops at :
(1) the red colour.
(2) the green colour.
(3) the yellow colour.
b Find the probability that the pointer does not stop at the red colour.
[30] In the arrow and target game , the target was on the form of a square
divided into the parts shown in the given figure and
the player was advised (asked) to throw the arrow at
(towards) the target without falling in an area on a certain
distance away from the target :
а Find the probability that the arrow hits area (D)
b Find the probability that the arrow hits area (A)
c Find the probability that the arrow hits the area (B) or (D)
[E The following table shows a sample formed from 200 TV viewers of TV
programs, they were asked about their prefered program :
Program Sports News Series Films Songs
Number of viewers 70 20 45 35 30
If a viewer is chosen at random , what is the probability that he is a viewer оғ?
a News. b Songs. с Sports. d Series. e Films.
(218
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى EE
Unit Four
(EB The marks for the students of a class in Math exam is recorded in
( the following table , if one of the students is chosen randomly.
Find the probability that this student got "good".
Excellent Very good Good | Раз5 Weak
8 12 te | 12 8
(In one of the classes of the sixth primary ; the teacher of Maths
classified the levels of the students in his subject into (weak - intermediate
- advanced) their number is 40 students and recorded his data in the
Number of
opposite table :
One of the students in this class is ihe level the students
chosen at random. Weak 5
Calculate the probability of : Intermediate 25
Advanced 10
The sum 40
a The event А: where A is a weak
student.
b The event B : where B is an advanced student.
€ The event C : where C is not an intermediate student.
| опе of those companies that produce fridges made a survey about its
production for a group consists of 500 women to get their opinions about
the sizes of these fridges ; the results were as follows :
12 14 16 Total
Size of the fridge in feet 6 10
Number of women 25 90 165 130 90 500
If a woman is chosen randomly; what is the probability that her fridge's
size is :
a 6 feet. b 10 feet.
(Ba class has 50 students. The number of girls is less than the number of
boys by 10 ; if one student is chosen randomly ; find the probability that
c 12 feet. d 14feet e 16 feet.
the student is a boy.
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى EE
Test on Unit
Answer the following questions :
п Choose the correct answer :
a The sum of measures of the accumulative angles about the centre of
a circle = : (90° or 180° or 270° or 360?)
A regular die is tossed once ; then the probability of appearance
ап even number = = (1 or Т or i or
The opposite figure represents
the percentages of distribution of
the sport activities for the pupils in Basketball
a class of a school , their number р КЦ
is 40 pupils , then the number of
pupils who participated in
basketball = ------- pupils. (20 or 12 or 8 or 5)
The measure of the central angle of the circular sector whose area
represents i from the area of the circle = =- R
(30 or 45 or 60 or 90)
e The probability of the impossible event = .......... (Ø or 1 or 0 or 2)
Complete each of the following :
а In the experiment of tossing a regular coin once , then the set of
the sample space (5) = =
b In the opposite figure :
(1) The percentage of the members who
prefer art is i %
(2) The measure of the central angle of
cultural = ------------
220
ر هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى Tus]
Unit Four
с Aclass of 50 pupils , if the probability of success for those pupils in the
end year exam is 0.8 , then the expected number for the pupils who will
succeed -
d If Sis the sample space of a random experiment . then P (S) = ----------
е Abasket contains 15 balls numbered from 1 to 15 » a ball is drawn
randomly ; then the probability that the drawn ball carries a number
divisible by 5 is
(Bie following table shows the percentages for chickens production in four
farms in a month :
Farm 4st | gnd | 3 4h
Percentage of production | 30% |
(1) Complete the table.
(2) Represent these data by a pie chart.
Ba box contains 4 white balls , З blue balls and 8 red balls › all of them are
symmetric ; a ball is selected without looking inside the box.
Find the probability that the selected ball is :
(1) Blue. (2) Not red.
(3) Green. (4) Blue or red.
(Be following table shows the number of studying hours that Tamer done
in a week :
Subject Arabic | Maths | Science | English | Social studies | Total
Number of hours 6 10 7 9 4 36
Represent these data by circular sectors.
(221)
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى EE
| Activity } Using Excel program to represent data by a pie chart.
Example &
The following table shows the percentages of time spent by Hassan
Lu ~ ©
W с> 9 2
=
studying different subjects during one week : 4n;
8
Subject Arabic | Maths | Science Art Social studies £
The percentage | 30% | 27 % 19% 7% 17% W
Use the “Excel program” to represent these data by using the circular sectors. JF
Procedures of activity : 6
[1] Click "start" جه “All programs" ___, select "Microsoft Excel". \ 1
[2] Write the data of the previous table as shown in figure (1)
BK - ١ Ф 2
2
"X23
D
A [se Ic] о T7 ع ШЫ )
Subject Arabic Maths Science Art Social studies А
The percentage 30% 27% 19% 7% 17% E
ڪڪ
9 %
9
V
el ?
bad
C3
Fig. (1)
$t C>
wq с> 9
أهذا العمل خاس بموقع p التعليى ولا يسمح بتد وله على مواقع أخرى M PERSE]
Iouem] GES) [a ae | \
Unit Four
(223)
"هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى Tues]
[5] From the dialogue box of "chart wizard - chart type" select "pie" as shown in
figure (4)
(224)
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى EE
Unit Four
Char Area
А. Н ишегим шы 7 LET F
Subject — [Arabic Maths Science Art Social studies
The percentage] 30% 27% 19% 7% 17%
FEER
БЕЗЕБ
Urn تيرم ؟ / uus V رياضيات لغات waked! 225,
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى EE
ty of Unit Four
а Ask your classmates about their times of getting up from the following times :
(Before 6 : 00 a.m. - at 6 : 00 a.m. - at 6 : 30 a.m. - at 7 : 00 a.m.)
Then tabulate the data you get in a simple frequency table and represent
these data by a pie chart using Excel program; then print the sheet.
@ 10 Toss a coin 30 times; record what you have got in the following table :
a Calculate the probability of
the event A where A is the event of
appearance of a head.
Calculate the probability of
the event B where B is the event of
appearance of a tail.
-i
The event
The tally
The
frequency
Head
Tail
The sum
30
с What is your expectation about the chance of appearing of each of the
head and the tail if the number of tossing times increases to be :
100 times - 500 times - 1000 times.
ДЕШЕ] هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى A
© О >
A "WW
A
2
$
S 4
6 A
EE هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى ٠
i 0
TIMSS“ Questions
Choose the correct answer :
1 4 СА
(а) 90 (b) 80 (c) 50 (d) 40
2 In which of the following are the angles ordered by measure , from
smallest to greatest ?
R
025 0265
(5 2.9 S.R.P.Q
3 Twice the number y subtracted from it 4 the symbolic expression for this
situation is с
a)y-4 b)2y-4 у+4 1) 2у+4
4 |f the pattern 3 , 6 „9 , 12 was continued , which of these numbers
would be one of the numbers in the pattern ?
)26 b) 27 c) 28 )29
ж TIMSS : Trends of the International Mathematics and Science Studies.
TIMSS Questions
How many lines of symmetry does
the opposite figure have ?
(a)4 (b) 3
(с)2 (d) 1
The smallest prime number is -- (El-Fayoum 2017)
(a) zero (b) 1 (d)3
How much do the apples weigh
in grams ?
(a) 200 (b) 202
(c) 210 (d) 220
In which number does 8 have the value of 800 ?
(a) 1468 (b) 2587 (c) 3809 (d) 8634
Which of the following figures the shaded area represents 2 of the square ?
(Cairo 2016)
AELIAN. NEF
10 Which fraction is not equal to the others ?
(а) i (b) $ () 2 (9 2
11 Mariam stacks these boxes in the corner of
the room , all the boxes are the same size.
How many boxes did she use ?
(a) 25 (b) 19
(c) 18 (d) 13
12 Salma paid L.E. x for buying three pens , then the price of each pen
(Et-Dakahlia 2016)
e ož ()3x (d)3+x
@29)
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى 152551
(D
13 The highest common factor for the two numbers 18 and 24 is
(a)6 (b)3 (c) 2 (d)4
14 Which number is 100 more than 5432 ?
(а) 6 432 (b) 5 532 (с) 5 442 (9) 5 433
15 3... {1 533 535} (Ismailia 2017)
(€ (b) (ос (e
16 The perimeter of a rectangle is 16 cm. ; its width is 3 cm. , then its
area = om? (Red Sea 2016)
(a) 15 (ط) 9 (с) 48 )0( 4
17 The angle between the two hands of the clock is right when the time is
. o'clock.
(a) 12 (b)6 (c)3 (d)2
18 54.76 =- (to the nearest tenth)
(a)50 (b) 55 (c) 54.7 (d) 54.8
19 The number of symmetry lines of the isosceles triangle = ~~
(Beni Suef 2016)
(a)3 (b)1 (c)2 (d) zero
20 All numbers are divisible by 2
(a) even (b) odd (c) prime (d) decimal
Second : Complete each of the following :
1 The smallest odd number is ·----
2 The side lengths of a triangle are 3 cm. . 4 cm. and 5 cm. ; then its
perimeter = ==- cm. (Et-Sharkia 2016)
3 The place value of the digit 5 in the number 256 374 is ---------
- = 8 253
5 Pentagon is a polygon of --
5-5
Tues] هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى
TIMSS Questions
In the opposite figure :
AD ل BC , then the length of AD is
called =- of A ABC (Damietta 2017)
If {3 ,X} = 15 ,3} , then X + 1 = --------
The lowest common multiple for the two numbers 10 and 15 is =-
Number
The opposite graph shows the
number of cartons of milk sold
each day of week at a school
» the number of cartons of milk
the school sold on Wednesday
à 8 i à
10 The number of lines of symmetry of the rhombus = (North Sinai 2017)
11 3 105 + = = 3
12 A prime number between 1 to 10 is (Souhag 2016)
13 One day and two hours = hours.
14 3.26 km. = m. (South Sinai 2017)
15 Six sevenths =
16 8 +8 + ق +8 = 8 х ----.-....
17 3.75 + 2,5 = --- zo (to the nearest 45 (Ei-Menia 2016)
18 ї 9
The opposite calendar for December DECEMBER
Jana's birthday is on Thursday د Su Tu We Th F
December 2 , she is going on a trip 1253
7 *& э то
exactly 3 weeks later , then she will 14 15 16 17
21 22 23 24
go on the trip on the date ~- 28 29 30 31
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى EE
TIMSS Questions
19 8925 z ......... (to the nearest tenth) «Aswan 2016»
201((2,3.,5) 0 {5,7,3} = 3] > then X= =-
Third : Answer the following questions :
1 In Sara's class , there are twice as many girls as boys ; there are 8 boys
in the class. What is the total number of boys and girls in the class ?
Bassem is arranging squares in the following way :
ч MEN
Figure 1 Figure 2 Figure 3
How many squares [ | would Bassem need to make figure 15 ?
Amgad ate i of a cake and Amal ate i of the cake. How much of the
cake did they eat altogether ?
In a football league , teams get :
3 points for a win , 1 point for a tie , 0 points for a loss
Stars team has 11 points , what is the smallest number of games Stars
team could have played ?
George practiced tennis six days a week.
For 3 of the days he practiced for 45 minutes each day.
For 3 of the days he practiced for 20 minutes each day.
In hours and minutes; what is the total amount of time George practiced
on these six days ?
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى EE
27 على مواقع أخرى Agl هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح d.
AD MARY
able to
absolute
according to
additive
advanced
afternoon
age
agree
always
apparent
appearance
appropriate
arc
area
associative
Atlantic
average
axes
axis
ЕВЕ FE 7 ALAL y A1
backward is
balance
balloon
base
begin
behind
bell
belong to
board لوح / لافتة / كرتون
طامط э
bound مرتبط
box
bulb بصلة
(234)
ينتمى إلى
calculator الحاسبة 2%!
cardboard مقوى dos / كرتون
carry يحمل
carton علبة كرتون
case Ue / ile
ceiling
cell
central
ceramic
certain
chart
check
circle
circular
closure
circumference
click
close
coin
coloured
commutative
compare
comparing
compasses
competition
complement
component
concept
consecutive
constant
container
coordinate NI
copy ينسخ / نسخة
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى EE
Glossary
correspond to
cost
cover
cube
cuboid
damage
data
decimal
decrease
deduce
deep
degree
denote
deposit
depression
depth
describe
description
design
determine
diagram
diameter
dice
die
difference
dimension
direction
directly
discover
displacement
distance
distribution
distributive
diver
dividend
diving
divisor
dot
double
down
drag
drawing
drop
during
edge
electric
electronic إلكترونى
element عنصر
elevation ارتفاع
energy طاقة
entrance دخول / مدخل
equality تساوى
equation
event
except
exchange
existence
expenditure
experiment
explain
exponent
express
expression
6 sti
ess
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى EE
Glossary
face
factory
farm
farther
favourite
few
fitness
fold
forecast
form
formula
forward
fridge >
gain
general
geometric
give
given
graph
graphically
grocery
imi ممت oR
hand
head
height
helicopter
high
hill
hit
hobby
horizontal
hundredth
Иш с See 1
їйеа! "n
identity
image
important
impossible
incline
include
increase
index
inequality
inner
inside
integer
interesting
intermeditate
internally
intersect
inverse
iron
kick
largest
lateral
lattice
layer
left
length
level
lid
lie
listing
loss خسارة
233
rei Y تيرم / uis V old رياضيات укай
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى EE
order
ordering
orthogonal
outcome
outer
magnitude
major
marble
maximum
measure
mention 4 painting طلاء / دهان
merchant Xt pair زوج
midnight > рап КИЗ:
тїпог 3 parallel يوازى / موازى
missing parallelogram متوازى الأضلاع
most participate يشارك
mountain "7 particular >
move pattern
movement peach
multiplicative ١ penalty
LE | percentage
natural = perform
nearest perimeter
necessary : pie charts
negative ы plane
neutral d pointer
news >i polar
next pool
non-negative position
non-positive positive
numerical possibility
possible
power
practice
preceding
predict
prelude
observe
occur
once
opinion
opposite
234
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسم بتداوله على مواقع أخرى آ22
prevent
previous
print
probability
product
production
profit
program
property
quantity 3
quotient خارج القسمة
3ك" سر Hp
أنصاف أقطار الدائرة radii
نصف قطر الدائرة radius
يرفع raise
random
reason
record
rectangle
reflection
relation
repeat
repeated
represent
required
respect to
respectively
rest
rhombus
right
rise
rotation
(238)
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى EE
round
row
rule
rust
دورة / حول
sample
satisfy
Scale
sector
select
semicircle
sentence
sequence
series
set
shaded
sheet
side
sign
simplest
simplify
situation
size
space
specialized
spinner
square
starting
statement
statistical
statistics
stick
store
string
submarine
subset
substitute
substitution
succession
suffer
suitable
sum
summarize
supply
sure
surface
survey
symbol
symmetric
symmetry
tail
tank
target
tart
technological
tick
tie
tile
tossing
total
trade
training
transform
transformation
translate
translation
236
uncoloured
unfold
unique
unknown
upper
usually
valley
value
variable
verify
vertex
vertical
vertices
wall
withdraw
withdrawal
without
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى EE
Е]
ts on unit (1) and unit(2)
on unit(3) and unit(4)
Worksheets
on unit (D and unit (2)
EEE العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى y.
Mb ححاب تمماصر] | Baan 3
On lesson 1 unit 1
E Complete each of the following :
[а] Z' N ZT = e [b] | - 131=
[c] The opposite of 7 is ===- [d] Z-N-
[e] NN Z = e
Ө Put the suitable sign "E, ¢ Cor": —
а-7 [ ] x m3 [ |z
ыт [|м 111-351 |w
ен [| |Z
Write an integer to represent cach situation :
[3] A temperature of 2 degrees below zero.
[b] An increase of L.E. 7
[c] 9 m. above the sea level.
[d] A loss of P.T. 80
[e] A bank deposit of L.E. 95
B rina the result of each of the following :
[а] 3 *|-3|*- [ь]125|+|-5|=
[e] |-11|+ 22 = . [9]1-61х131=
[е11-91х0= ~
Represent the following numbers оп the number line :
[a] -3 [511-51
[е] - 121 га {7,8,9}
هذا العمل خاص بموقع ذاكرولى التعليمى ولا بسمح بتداوله على مواقع أخرى [стз
[1] Put the suitable relation "> , = or <":
[a] – 8 4 [b] 0
]| [
ا
[c] 5 1-51 [d -3
Ie]l - 91 -|-10|
[a] Arrange the following numbers in an ascending order :
—6515,0;|-9|and—- 18
[b] Arrange the following numbers in a descending order :
—9 -|و 17 و 9 | »— 15 and 16
Complete each of the following :
[a] The number ~- is neither positive nor negative.
[bD]- 45-35-22 55 (in the same pattern)
[e]Z-Z-=
[d] The smallest positive integer is
Ie]l - 12] * 1-21] 7 -
Write :
[a] The previous integer and the next integer of — 27
[b] The integers between the two integers — 5 and 3
B write using the listing method each of the following sets :
[a] The set of integers greater than — 4
[b] The set of integers smaller than — 1
[c] The set of non-negative integers.
[d] The set of integers smaller than 5 and greater than — 6
TEE] هذا العمل خاص بموقع ذاكرولى التعليمى ولا بسمح بتداوله على مواقع أخرى A
From Lesson 1 unit 1
to lesson 3 unit 1
Find the result of each of the following :
[а] (-7)+2=-
[b] )- 4) + (-5) =
[c] 14-27=
[9] 16 - (- 3) =
BITTE E S B
B write the property of addition in Z in each of the following :
[a] (- 5 + 6) + 9= =5 + (6 + 9)
[Ы (= 8) + 7 = 7 + (-8)
[ce] 11 +0 = -1
[d] 14 + (- 14) =0
B Complete each of the following :
[a] 4 + (-7)-2=
[b]-8+5+8=-
fe] |-7|+
[e] The additive inverse of (- 5) is ----------
[а] Use the properties of addition іп Z to find :
[a] - 15 + 29 + 15
[b] 55 + (- 255) + 45 + 255
Arrange each of the following in an ascending order :
[а] 131,5 ,-3 ,„Оапа4
[b] 1,11, 1and- 41
ر هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى Toss]
From Lesson 1 unit 1
to lesson 4 unit 1
(E Complete :
[a] The product of two negative integers is
[b] 17 x = = (-5)x 17
[c] |- 2 | = ———
Id Z* U {O} = ——
[e] 5 х (= 2) = eee
Find the result of each of the following :
[a] 9 x (- 3) [b] (- 36) + (- 4)
[с] [8 + 5)] x6 [4]6х[-2+(—7)]
[a] Use the properties of multiplication of integers to find each ©
of the following :
(1)50 x 14 x 2 (28x(-9)x125x3
[b] Use the distributive property to find the result of each of the following :
(1)3x(-2)*3x5
(2) 112 x 98 + 112 x (– 97)
Use the distributive property to find :
[a] 54 x 101
[b] 73 x 99
[a] Write using the listing method each of the following sets :
(1) The set of integers greater than — 3
(2) The set of integers included between — 4 and 2
[b] Use the properties of addition in Z to find :
(1)5 +4 +(—5) (2)45 + 36 + 55 + 64
Qo Yes {әх | (Worksheets & Examinations) رياضيات зіч! @)
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى [сте
Sheet Ө |
From Lesson 1 unit 1
. to lesson 5 unit 1 "P --
1] Choose the correct answer :
[a] - 7) (€ or € or С or ¢)
[b] The additive inverse of (- 2 is (9 or З or —3 or -9)
[e] (— 9)2 = sss... (—81 or —18 or 81 or 18)
[d] f| - 4 | 2 X then x= б (4 or –4 ог 16 or – 18)
[e] #-7 +п= —7 > then - (1 ог 7 or -7 or 0)
Find the value of each of the following :
la] 54 х 53 = للق.......
[b] 99 + 97 =...
[c] 8? x 8 x 8?
cT.
c7
Simplify each of the following :
Idi
(p Se
a Put the suitable relation "> ; = or <":
[a] - 12 - T
Ib] c 1)
[c] = لعي ( 10e"
Id) |-6 |+ 5)?
А табора)
апа 2 1(15 -) و *)3 -) و 4)0 -) و 29 -(
هذا العمل خاص بموقع ذاكرولى التعليمى ولا بسمح بتداوله على مواقع أخرى TEE]
[1] Complete іп the same pattern :
[31256 10, 14, Е
[b] 154.9;
CEEE ЖЕ EAC
[d] 10 000 و 1 000 , 100 ;-
[e] -2 s032 yÅ 5 08
Describe the pattern , then complete in the same pattern :
[a]5 » 13.2129 ,
[b] 25 , 21,17,13 ›-
[c] 1 25458516 5 зс
Look at the pattern of dots د then answer :
[a] Draw the 4* and the 5' shapes.
[b] How many dots will be there in the 4'^ and the 5th shapes ?
[а] Arrange each of the following numbers іп an ascending order :
[a] )- 4)? »-5.|- 51.056 and - 14
Ib] 11:50 ,-|7].— 11 +0 and 3?
Choose the correct answer :
(€ or є or C or $)
[b] If mx 7 = 0 » then m = ------------ (1 or 0 or 2 or -7)
[c] 8 + ~- -2 (6 or —10 or 10 or -6(
[d NUZ- = (Z or Z* or {0} or N)
[e] 6-(-9) = (-3 or 3 or 15 or – 15)
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى EE
[1] Find the solution set of each of the following equations :
[a] X + 5 = 12 sif the substitution setis: {3 55 58 57}
[b] 3 X—4 = 8 , if the substitution set is : {3 5 26}
[c] 2 X + 1 = X —8 , if the substitution set is: (2 »4 ;— 1 »- 4]
[d] 3 (X — 2) =- 6 » if the substitution set is : {— 1 0 >t}
B Find the solution set of each of the following inequalities :
[a] 3 X + 5 > 2 , if the substitution setis: {-2 »— 1,0,1}
[b] 3 X— 1 >—2 > if the substitution setis: [-2 »—1 031 2}
[c] 5 X— 1 > 4 sif the substitution setis: (2 3 4.5 +6}
[d] X + 3 > 5 » if the substitution set is: [051 2 ,3,4}
Considering the set of substitution is A = (0 +1 +2 3}
Find the solution set of each of the following :
[a]2X-7 2-1 [b] Х+4>5
[а] Complete :
[a] The additive inverse of — 4 = ===
[b] | -9 | + 3 = 0
[c] 15254585 16 5 2
Id 2* N ZT = -----
[e] The multiplicative neutral element in Z is
a [a] Simplify : s х 5
[b] Determine the degree of each of the following equations :
(1)-4b= В — — (2yx3— 9X2 4
eys (4) x^ «3 x5- 9
©
ر هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى EE
(Ва the solution set of each of the following equations іп N :
[al2x-1=9 [b]j3X*2-7 17
[c]3x-4=11 [d]4x-3=-7
(rina the solution set of each of the following equations in Z :
[alx+8=-3 [b]3 X * 2 » — 19
[c]2X * 42-4 [d]2 X * 1 = 13
[Еўсотрее #
[a] The degree of the equation : 3 X? + 4 х—1 = 0 is -
[b] If | X| > 7 then X = ог = ممت
[o]Z* 0 ZT = eee
[d] The number is neither positive nor negative.
[e]3 59 , 27.81 ; (in the same pattern)
[a] Use the properties of addition in Z to find :
(1)25 + 13 + (- 25)
(2)5 + (-3)+7+(-9)
[b] Arrange ascendingly :
-5 5|-5| (-2)} , 0 and - (3?
[a] Use the multiplication properties of integers to find :
(1) 50 х (-31) x2
(2) (— 25) x 9 x (- 4)
211
b] Simplify :
[bl Simplify : 2—2
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى EE
[1] Find the solution set of each of the following inequalities :
[a] 2X + 1 > 7 » where XEN
[b] 2X- 325 where X EZ
[c] 3 X + 1 = 13 , where XEN
[d]-2X»5.whereXCZ
B Find the solution set of the inequality X + 2 < § where :
[a] X€N
[b] XEZ
» then represent the solution set on the number line.
Use the distributive property to find the result of each of the
following :
[a] 23 x (- 121) + 23 x 21
[b] (= 35) x (- 72) + (- 35) x 2
[а] Complete :
[а] The degree of ће equation: 2 X + 1 = 5 is م
[b]The additive inverse of (— 8 is =
[c] 35 + 35 = ..........
[d] The greatest negative integer is =-=-
[e] 27 ١ و1 c (in the same pattern)
Use the multiplication and addition properties of integers to find :
[b] - 15 + 15 + 9
)14(
٠ هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى [ЖЕЕ]
Worksheets
on unit (3) and unit (4)
]هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى Tess]
ema E | ححاب تمماصر] Mb
A(15—1) و B (4 ,-1) and C (4 , 5) » then find :
[a] The length of each of AB and BC
[b] The type of the triangle ABC with respect to its side lengths and its angles.
[c] The area of the triangle ABC
[1] Determine the position of each of the following points
(Bine opposite figure : -
ABCD is a rhombus ; complete :
[b] The length of AC =
[c] The length of BD
[d] The surface area of the rhombus ABCD = ~=. "A
Determine the positions of X (2:2) » Y (-25—-3) › Z(3.-3)
and L (3 , 2) » then find :
[a] The name of the shape XYZL
[b] The perimeter and the area of the shape XYZL
[c] The number of axes of symmetry for the shape XYZL
(Bl Determine the positions of L (—2 ,- 1) > M(1s-1) > N(153) ©
and P (- 2 »3) » then find :
[a] The length of each of LP and PN
[b] The perimeter and the area of the shape LMNP
(El on a square lattice , draw A QRS where Q 1.73) > (3-3) га 3
and S (1 , 6) » then find :
[a] The length of QR
| TRe"typerof‘thertriangle"@RSaccording"torits"side"lengths?
[c] The number of axes of symmetry for the triangle QRS
: )©
| هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى |1822
_ From Lesson 1 unit 3
to lesson 2 unit 3
Bon a square lattice د draw A ABC where A (5 , 3) د В (1 , 1) апа
C (6 »— 3) د then find its image by translation
(х,у) —» (X-4 »y +1)
On the coordinate plane ; determine the points A (1 52) » B(-2;2)
and C (- 2 ,— 4) » then find :
[a] The length of AB
[b] The length of BC
[c] The image of A ABC by translation (3 »— 1)
If A (1 »—1) and B (- 1 » 3) » write the mapping rule of the translation
that makes B the image of A
Go a lattice د plot the vertices of the triangle ABC where
A(151) و B(-3 ›– 1) and C (0 ,—5) » then draw its image
by translation (X + 4 ,у + 1)
On a square lattice ; draw A MNT where М (1 ,– 3) و N(-3;1)
and T (— 2 ,— 5) و then draw its image by translation of magnitude
3 units in the positive direction of y-axis.
(Yee Гол | (Worksheets & Examinations) رياضيات жача 62
هذا العمل خاص بموقع ذاكرولى التعليمى ولا بسمح بتداوله على مواقع أخرى [стз
Lm
«o9 me,
From lesson 1 unit З т
to lesson 3 unit 3
In the opposite figure :
A circle M of radius 10 cm. is divided
into 8 equal circular sectors.
Calculate the area of one sector (consider Jt = 3.14)
(8) If the length of the diameter of a circle is 14 cm. Calculate :
(1) The circumference of the circle.
(2) The surface area of the circle. (consider 7t = 22)
(b) Find the area of the opposite figure
(consider xt = 22
я In the opposite figure :
Find the area of the shaded part. (consider 7t 7 3.14)
(8) Determine in the coordinates plane the image of the line segment
AB where A (2 +3) و 8 )- 2 0) by translation (X + 3 ,у- 2)
(b)A circle ; its circumference is 88 cm.
Calculate its radius length and its surface area. (consider w= 22 )
ر هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى Tus]
From lesson Ї unit 3
to lesson 4 unit 3
Ю @ A cube-shaped box is of edge length 5 cm. Find :
(1) Its lateral area.
(2) Its total area.
(b) A cuboid is with length 7 cm. » width 5 cm. and height 8 cm. Find :
(1) Its lateral area.
(2) Its total area.
(а) If the sum of the edges of a cube is 108 cm.
Find its lateral and total area.
(b) A cuboid is with square base of side length 3 cm. and height 6 cm.
Find its lateral area and total area.
5 The perimeter of the base of a couboid is 20 cm. and its height is 6 cm.
Calculate the lateral area of the cuboid.
Ф If the lateal area of a cube is 100 cm? Find its total area.
@ A cuboid with a square base whose perimeter is 20 cm.
and its height is 8 cm. Find :
(1) The lateral area.
(2) The length of its base side.
(3) The total area.
(b) Find the area for the circle with diameter length 14 cm. (consider = 22)
В @ A cuboid whose total area = 132 cm? and its lateral area = 112 cm? a
Find the area of its base.
(B) A cuboid whose lateral area 140 cm.? and the dimensions of its base are
6 cm. and 4 cm. Find its height.
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى EE
lunt4á — —
)1[ The following table shows the rate of the score of 300 pupils
in one school :
Rate | Excellent | _ Good Pass Weak
Percentage | 20% | 45% 25 96 40 96
[a] Represent these data by a pie chart.
[b] Find-the number-of-excellent-pupils.- —
B8 The following table shows the percentage of the time Ayman
spent studying some subjects during a week :
[ Subject | Arabic | English
[ Percentage | 30% |
[a] Complete the table.
[b] Represent these data by a pie chart.
B The monthly income of a family is L.E. 1800 , the family spends 25 % of its G2
income on rent » 40 % on food » 20 % on others and saves the rest. NS
[a] Represent these data using the circular sectors.
[b] Find the capital which this family saves monthly.
[а] The following table shows the number of studying hours that ^W
Mohamed has done in a week :
| Subject Arabic | Maths | Science | English | Social studies | Total
[Number ofhours| 9 10 6 z 1] 4 |] 36
Represent these data by a pie chart.
( [EJ tal A cuboid › its length is З cm. د its width is 2 cm. and its height is 4 cm. G
Find its total area. 5
{b]-A:diameter-length of-a-circle.is-20.cm:
Calculate its surface area. (Consider w= 3.14)
)85(
| هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى [ЖЕЕ]
From lesson 1 unit 3 25
to lesson 2 unit 4
[1] A bag contains 8 equal cards د have the same colour د numbered from
1 to 8 Write the sample space for this experiment.
(E From the set of digits {1 »5 , 7} د а number is formed from two digits د
determine the sample space of this experiment showing the number of its
elements.
Determine the sample space of tossing three distinct coins once
and observing the sequence of appearance of heads and tails.
[а] [а] In the coordinates plane د draw the rectangle ABCD where ©
А(4,2) » B(4:4) » С(1,4) and D(1 ,2) sthen:
(1) Draw its image by the translation (X + 2 »y + 2)
(2) Calculate the perimeter of the image of the rectangle ABCD
Ib] Find the area of a circle with diameter length 28 cm. (Consider zt = 22)
B [a] Find the total area of a cuboid with square base of side length 6 cm.
and height 8 cm.
[b] The following table shows the percentage of production of electric sets :
Sst | + zd | ss 4^ |
Percentage | 40 96 1596 | 30% 15% J
Represent these data by a pie chart.
[стз هذا العمل خاص بموقع ذاكرولى التعليمى ولا بسمح بتداوله على مواقع أخرى A
From Lesson 1 unit 3
to lesson 3 unit 4
[1] А bag contains 15 cards numbered from 1 to 15 , if one of the cards is
chosen randomly ; write the sample space for this experiment and the
number of its elements ; then find the probability that the chosen card :
[a] Carried an even number.
[b] Carried a number divisible by 4
[c] Carried a number satisfying the inequality : 3 s X 1 «9
B A bag contains 25 balls (4 balls are yellow د 9 balls are red and the
remainder is black) د if a ball is drawn randomly ; find the probability
that the drawn ball is :
[a] black. [b] yellow or black. [c] not black. [d] brown.
A bag contains 20 similar marbles. Tarek drew a marble randomly e
4
and he found it red. If the probability of drawing а red marble = $
Find the number of red marbles in the bag.
(EE [a] The спо of a cuboid is 3 cm. » its width is 2 cm. and its height is 4 от. (С
Find the total surface area of the cuboid.
[b] In the opposite figure :
A circle of radius length 7 cm. is divided Va
into 8 equal circular sectors. Find : D ANS
(1) The area of one circular sectors. (Consider л =22 ) NS
(2) The measure of the central angle of sector.
a] If the perimeter of one face of a cube is 20 cm.
Find its lateral and total area.
[b] The following table shows the ratio for producing chickens in
four farms in a month :
Farm 4th
The ratio of production
Represent-these-data-by-a-pie chart.
)@
"هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى EE
BED Ae التصل
SUMMARY
OF THE SECOND TERM
(— — 4 Summary of Unit One y
The set of integers "Z" - {- 26,6 2 = I „О, 22459.)
It ie formed from the union of three sets Z , {O} and z*
ie. Z ={-,-8,-2,-1,0,1,2,3,-}
J | E 9
" U (o) U а a8
e0Z*' and غ06 z-
+ The set of non-negative integers = {0 » 1,2, ..} = {0} U Z*=N
• The set of non-positive integers = 10 -و 1 »— 2 »= 35 -—-}={0}U Z7
e The set of odd integers = {... 3-و ,—1 5153 .--
• The set of even integers = {... [...و 032,4 2—» 4 دو
Representation of the integers on the number line
Positive integers “Z+ "
are the right of O
Negative integers “ Z7”
are the left of O ighe тор
5 4 3
[ Opposites (inverses) and absolute value
On the number line , any two numbers that are at the same distance from O
and on two opposite positions of it are called opposites or inverses.
For example :
; 5 unite : 5 unite
-6 -4 =3 2 -[ © t
i.e. The opposite of 5is- 5 апі
е)
EE "هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى
Summary
The absolute value
The absolute value of a number is its distance from О on the number line.
The absolute value of any number X ie denoted by | X |
The absolute value of any number (except О) ie always positive.
Examples :* | 41= 4 1-41=4 О-О
| Ordering and comparing integers
e For any two integers a and b» if the point representing a is to the left of the
point representing b »then a « b
For example :
-4<-1
а b
-5 €4-3 -2 С) о |
because the point representing — 4 lies on the left of the point representing — 1
Any positive integer is greater than any negative integer.
e Zero is smaller than any positive integer and is greater than any negative integer.
e The least positive integer is “1” and we cannot determine the greatest
positive integer.
• The greatest negative integer is “— 1" and we cannot determine the least
negative integer.
Ca
GEER audition
o To add integers have the same sign د keep the same sign and add the
absolute value of each number.
For example: ٠ 5 +4 = 9 ۰)-5( + )-4( = -
© To add integers with different signs د keep the sign of the number with the largest
absolute value and subtract the smallest absolute value from the largest.
For example : • (-8)+6=-—2 (because |—8|>|6 |)
*9+(-3)=6 (because|9|»|—- 3l)
)4: زم Yes [os | (Worksheets & Examinations) رياضيات walsell (55)
TEE] هذا العمل خاص بموقع ذاكرولى التعليمى ولا بسمح بتداوله على مواقع أخرى A
Summary
e The division is not always possible іп 2,
е The quotient of two integers with the same sign is positive.
Le. (9 + © -)© and © -© - ©
The quotient of two integers with different signs is negative.
Le. «Q-O аа О+@®=©
e The quotient of zero divided by any non-zero integer is zero.
e Division by zero has no meaning.
e The division operation in Z is not commutative.
• The divison operation іп Z is not associalive.
١ Repeated multiplication
e Ifa is an integer and n GZ* د then a x a x a x ... to n times = a" where a is
called the base and n is called the power » index or exponent.
Power جسم
3x3x3x3 =3
Base
Any number to the first power is that number itseif.
For example : «9! - 9 e(-3)'=-3
• Any number except 0 to the zero power is 1
For example : ه 5? = 1 e(-7921
e If the base is one and n CZ » then 1^ = 1
For example : • 15 = 1 6172 =1
For example :
(а)" if nis even
elf a €Zandn €Z* » then (- а)" = s
- (а) ifnisodd
i.e. • А negative integer raised to the power of an even integer gives
a positive integer.
A: negative integer raised tothe-power-of-an-odd.integer-gives
a negative integer.
Tess] هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى В
Summary
Rules of power
Rule 1
Ifa€z-(0) ın €Z*,m EZ” , then: | a" xa"=a"*” |
For example :
„32 × 3? = 32*3 = 35 = 243 oa? x a5 = а3+*5 = аё
If a is an integer and a #0 ,n €Z* , тЄ2* ,mz n , then:
For example :
٠25 + 22 = 25-2 = 23 =8
| Numerical patter
e Numerical pattern is a sequence of numbers according to a particular rule.
Describing of the pattern is discovering the rule of the pattern and expressing it
in words.
For example :
*3 *3 +3 +3
> 4 175. По 07135016
Description of the pattern
Each number ig more than ite preceding by 3
x2 x2 x2 x2 x2
" s ml am ал ى
3 > 6 » 12 » 24 » 48 » 96
Description of the pattern
Each number ie twice of ite preceding.
TEE] هذا العمل خاص بموقع ذاكرولى التعليمى ولا بسمح بتداوله على مواقع أخرى A
A Summary of Unit Two
It is a mathematical statement that has two experssions separated by an equal sign.
One (or both) of the expressions contains one unknown (or more).
For example :
X + 5 =—7 is an equation د the letter "X" is called the unknown or the variable.
The degree of an equation
It is determined by the highest power of the unknown (symbol) in the equation.
For example :
+3 х— 1 = 815 an equation of the first degree in one unknown X
+a? —2b- 10 = 0 is an equation of the second degree in two unknowns a and b
Solving first degree equation in one unknown
The solution of the equation is the number which satisfies the equation.
i.e. which makes the two sides of the equation equal.
[Example )- ا
Find the solution set of the equation :
xX + 4 = 6 if the substituion setis {—3 51 » 2}
Solution
Substitute in the left hand side of the equation for X by the elements of the
substitution set as follows :
Whenx--3
.. The left hand side = - 3 + + - 1 #6
When X = 1
^. The left hand side = 1 + 4 = 5 66
When عد = 2
2. The left hand side = 2 + 4 = 6
-. The solution set of the equation is 12
ر هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى Tease]
Summary
Find the solution set of the equation :
3x-1=-7 > where X GZ
Solution
v3x-1-2-7 (Adding 1 to each of the two sides)
..3X-1+1=-7+1
..3X=-6 (Dividing each of the two sides by 3)
=6 „х= 5 =
‚== ب X=-2E% .. The S.S. = (- 2)
| The inequality
It is a mathematical statement that has two expressions separated by an inequality
sign (< or >). One (or both) of the expressions contains one unknown (or more).
For example :
°2Х+1>7 ٠3-8
| The degree of an inequality
It is determined by the highest power of the unknown (symbol) in the inequality.
For example :
2 X+ 1 > - 8 is an inequality of the first degree in one unkown X
| Solving first degree inequality їп one unknown
Example )
Find the solution set of the inequality :
х- 3 > 1 if the substituion set is {6 5554 53}
Solution
When X = 6
г. The left hand side = 6 — 3 = З is greater than 1
When x-5
2. The left hand side = 5 — 3 = 2 is greater than 1
G)
EEE] هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى Й
Summary
When х=4
2. The left hand side = 4 — 3 = 1 is not greater than 1
When X= 3
-. The left hand side = 3 — 3 = 0 is not greater than 1
-. The solution set = (6 , 5}
Find the solution set of each of the following inequalities :
@2x-325 > where X EZ
@1-2x>-7 » where XEN
@-5<3x+1s10 „where X EZ
Solution
[О] v2X-325 (Adding 3 to each of the two sides)
22х-3+325+3
S AREG (Dividing each of the two sides by 2)
2X ?
ene e X24
.. The S.S. = {4555657 >+}
v1-2X»-7 (Subtracting 1 from each of the two sides)
41-2x-1»-7-1
.-2Х>-8 (Dividing each of the two sides by — 2)
.22X.-8
BIA es
2х<4 2 The 8.8.={3,2›,1›,0}
v-5«83x-*1s10 (Subtracting 1 from each of the three sides)
4-5-1«3x*1-1510—-1
0-6 > 3 9 (Dividing each of the three sides by 3)
9 <3%< 5ت .
“а 3 8
3 {TOTTI}
Summary of Unit Three y
| The distance between two points on the number line
The distance between two points on the number line = | Number of the ending point
— number of the starting point |
For example :
In the opposite figur- :
MN =|2-(—1)|=12 + 1 |= 3 units.
Graphing points in the coordinate plane
The position of any point in
the coordinate plane is determined
by a unique ordered pair
as in the opposite figure.
i
The distance between two points in the coordinate plane
For example :
In the opposite figure :
+ AB I xx
2 AB=|B-Al=|2-(-3)]
=12+3|= 5 units.
“CDi yy
А CD=|D-C|=|-4-2]
=|— 6 |= 6 units.
| Geometric transformations
There are three types of the geometric transformations which are shown in the
following diagram :
(9: ei | حب | (Worksheets & Examinations) رياضيات yalsdl (эз)
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى Tess]
Summary
The geometric transformations
- ( Reflection )-. "| Translation B )م Rotation. J-.
The translation is a geometric transformation which slides a shape from a place to
another place (image) such as every point of the original shape moves the same
distance in the same direction to form the image.
| First | Translation in the plane
с Finding the image of a point by a given translation
А is the image of A by translation of
magnitude MN in the direction of MN
Ga Finding the image of a line segment by a given translation
= M
AB is the image of AB by translation of
magnitude MN in the direction MN
Check that : AB = АВ and AB // АВ ]
The opposile figure shows the image of
a triangle by a certain translation.
Every point of the shape must move :
• The same distance.
* In the same direction.
36(
ر هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى Toss]
Original shape
Summary
( e Translation in the coordinates plane
The image of the point (х » y) C4 translation (a »b) | the point A (X + a sy +b)
Example
On a square lattice , draw A ABC where :
А )-3 1) و 8 (0 »5) and C (2 د 3) , then find its image by the translation
(х,у) جه (X + 4 » y — 2) "The translation (4 »— 2)"
A(-35,1) — A(-3+4,1-2)
Le. A (15-1)
B(0,5) جه В (0+4,5-2)
i.e. B (4 ,3)
C(2.3) —- 6(2*4,3-2)
їе. С (6 » 1)
A ABC is the image of A ABC
by the translation (X » y) — (X + 4 ,у-– 2)
| Area of the circle
The area of the circle = tr? where 7 = 22 ~3.14
| Lateral area and total area for each of the cube and the cuboi
Solid Lateral area (L.A.) Total area (T.A.)
Area of one face x 4 Area of one face x 6
Cube = Edge length x itself x 4 | = Edge length x itself x 6
i Lateral area * 2 x
Cuboid Peri i
ه0101 سے of Бана НӘШ (the area of the base)
e If a cube without a lid » then :
The total area = The area of one face x 5 = Edge length x itself x 5
e If a cuboid without a lid » then :
The total area = The lateral area + the area of one base.
© Ҝ
[её] "هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى
| Representing the statistical data by using the circular sectors
e Each circular sector has an angle whose
vertex is the centre of the circle which
is called a central angle.
The sum of the measures of the angles accumulating Central
around at a point as the centre of the circle is equal to 360° angle
Example )
The following table shows the percentage of the production of a factory of
house electrical sets :
The kind of set | Washing machine | Heater Oven Mixer
ercentage 20% 15% | 40% | 25%
Represent these data by circular sectors "pie chart".
Solution )
The measure of the central angle ca
i =20 °=72°
of washing machine = 455 x 360° = 72 Va
The measure of the central angle of heater = 45. х 360° = 54° a T
The measure of the central angle of oven = £ x 360° = 144°
The measure of the central angle of mixer = E x 360° = 90°
| Sample space "outcomes space"
It is the set of all possible outcomes for a random experiment.
It is usually denoted by the symbol (S) and the number of all elements of the
sample space is denoted by n (S).
Write the sample space of each of the following random experiments and
give the number of its elements :
© Rolling a die once and observing the number appearing on the upper face.
@)-Choosing.a.prime.number-less.than. 20.
© Tossing two distinct coins once.
6
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى Tess]
Summary
Solution CIEN E
(0s-2(1,2,3.45,5.6) ,n(S)=6
Q@s= {2535557511 513517 19) »n(S)=8 Ae
@s = {HH HT TH TT} »n(S)-4 oe
а
| Proba y of occurrence of an event
ie number of elements of
The number of elements of S
• The impossible event = Ø while the probability of the impossible event = 0 i.e. P (2) = 0
e The certain event = S while the probability of the certain event = 1 Ї.@. P (S) = 1
• The probability of the possible event = proper fraction.
e For any event A د we found that : 0 S P (A) S 1
e The sum of probabilities of all outcomes of a random experiment = 1
• If the probability of occurrence of an event A is P (A) » then the probability that it
doesn't occur = 1 — P (A)
25 cards are numbered from 1 to 25 › a card is drawn at random ; find the
probability that :
@ The number is even. © The number is divisible by 5
G The number is less than 26 © The number is 30
Solution )
© The probability that the number is even = E
( © The probability that the number is divisible by 5 =
© The probability that the number is less than 26 = (Sure event)
@ The probability that the number is 30 = Ê = 0 (Impossible event)
62
Й هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى Tess]
FINAL EXAMINATIONS 1
v
2 /
A © Model Examinations of the School Book P
1 (2 models + model for the special needs students)
È \
7 © 20 Schools’ Examinations from Different Governorates. 1
25 هم
e.
ا 9
5 4
ez К
هذا العمل خاص بموقع ذاكرولى التعايمى ولا بسح بتداوله على مواقع u (EELA
LI m] Gaara \
Answer the following questions :
[1] Choose the correct answer from those given :
(1)Cc 094 ( 1) E er (zero or —1 or 1 or 2)
(2) The image of the point (— 3 › 4) by translation (x » y — 4) is .
))-3.0( or (-7>4) or (-3.8) or (-1.4))
(€ or € or c or z)
(4) When tossing a die once ; then probability of getting a number on the
upper face more than 6 = + (2 or zero or 4 or 1)
Complete the following :
(2) #х+6=2 , x€Z , then X = verse
( 3 ) In the opposite figure :
ABCD is a rectangle
»then the area of A ABC
EM cm? B
(4) A box contains 5 white balls , 3 blue balls and 8 red balls all of them
are symmetric. One ball is drawn from the box at random. Then the
probability that the drawn ball is red = 00-00
A
0
[a] Find the result of : 4 х 32 + 32-73
[b] Find the solution set of the inequality: X— 223 و xe€z
height is 7 cm. Calculate the lateral area.
[b] The circumference of a circle is 88 cm. Calculate its area.
"هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى TEE
Final Examinations
( [a] Find the solution set of the equation: 3 Xx 9-3 .و xEZ
[b] The following table shows the percentage of the production of
а factory of house electrical sets :
[The kind of set | Washig machine | Heater | Oven | Mixer
[The percentage 30 96 | 15% | 40% | 15%
Represent these data by circular sectors.
6 Model
Answer the following questions :
E Choose the correct answer from those given >
(1) If2 x=—6 , then X E (М or @ or ZF or Z)
(2) The circumference of the circle = -~
(г or 2r or г? or r+2)
(3) When tossing a die once ; then the probability of getting the number 5
( zero or $ or 5 or 1)
(4) The number which satisfies the inequality : x > — 2 is 171-
(—1 or —2 or -3 or -4)
[2] Complete the following :
(1) 252 =
(2) The set of counting numbers (C)
(3)A cube of total area 150 cm? , then the length of its edge is
(4) In a 6" primary class , the marks of the students are given in the
following table :
Excellent | Very good Good Weak |
8 18 16 6 |
If one of students is эле chosen ; then the probability that this
pupil got good degree is ٠ “
B [a] Find the result оѓ: 6 x — 5 – (2 x 3) + 3
[b]-Find-the-solution-set of-the-inequality--x — 2 23 where oc EL
»then represent it on the number line. 4
"هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى EE
Final Examinations
a [a] Find the solution set of the equation : 2 x + 9 = 5 , where x cz
[b] In the opposite figure : A
ABCD is a rectangle where its length = 8 cm.
and its width = 7 cm.
Bom.
Н
Calculate the area of shaded part. B
[a] In a Cartesian coordinates plane د locate the points А (2 , 3) , B (4 , 3)
and C (4 »7) » then find :
(1) The length of BC
( 2) The image of A ABC by translation (0 , — 4)
[b] The following table shows the number of students partcipating in the
school activties :
Represent these data by circular sectors.
(1:) ابتدائي/تيرم ؟ s / (Worksheets & Examinations) sw صا رياضيات |!
TEE] هذا العمل خاص بموقع ذاكرولى التعليمى ولا بسمح بتداوله على مواقع أخرى A
Final Examinations
| Model examination for the special needs students
Answer the following questions :
[1] Complete the following :
(1)1312 =
(2) The probability of the impossible event = ===
(3)Ifx*223,x€N,thenx-
(4) The perimeter of the base of a cuboid is 10 cm. د its height is 4 cm.
» then its lateral area = ~- cm
B Choose the correct answer from those given :
(1)25 x 22 =... (27 ог 4' or 1)
( 2) The surface area of a circle = U x ===- (r or г or 2r)
(3)z* U {O} = ee (Z or N or Z)
(4 ) When tossing a fair die once , then the probability of getting an odd
number = -------------.. ( t or i or 4 )
(EB Put true (и) or false (x) :
(1)|-5|+5=10
(2)If3x=9,thenx=-3
(3) The probability of the sure event = zero
B
(4 ) In the opposite figure : E m
The distance between the points A and B - 2 units.
TEE] هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى A
Final Examinations
Join from column (A) to column (B) :
A
(1) The sum of the measures of the angles of the
sectors about the centre of the circle = --------------
(2)2-——*
(3) The solution set of the inequality : X +2 < 5 ,
where x EN is
(4) The image of the point (3 , 2) by transtation
(152)is
[а] Complete the following :
The length of the edge of a cube is 4 cm. Calculate its total area and
lateral area :
The total area > 6 x
The lateral area = 4 x vee
2° x (-2)
[b] Find the result of : 25
TEE هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى А
Model Examinations
Answer the following questions :
C Choose the correct answer :
(1) А fair die is thrown once د then the probability of appearing me потег
3 equals eo or Ра )
(2) The solution set of the equation : 2 X — 6 in Nis ----
(1-3) or (3) or (2) or 2)
(3) (I-13]] 2 (€ or @ or C or 9)
(4) Ifx+522,thenxe (3 ог -3 or 7 or -7)
(5) The integer that lies between — 4 and — 1 is eM
(=2 or -5 or 3 or -4)
(6) (5)? х Qe (10° or 10 or (10)? or (10)
(7 ) If Ais an event in a sample space S , P (A) = 1 » then Ais event.
(impossible or simple or sure or independent)
(El complete each of the following :
(0)Z'-z-w--
(2) 14 + 213 + (- 14)
(3) The sum of PNE tom. of a cube is 84 cm. د then its lateral area
equals ~~ - cm?
(4) The image of the iod (2 »— 1) by translation 3 units in the positive direction
of y-axis is -. ^
(5)Ifx*6- Suis SEE siet xn
(6) (4х3 +3) – (7 х3) = eree
(7)IfX2|-3| و у=-2,ћеп2ху= ---.
(8) i з 2 345. 4 diu (in the same pattern)
Choose the correct answer :
(1) The multiplicative identity element іп Z is ....
(-1 or 1 or 0 or 2)
(DIZNE =~ ({0} or @ or z or zero)
(3) The surface area of the circle = ----
"هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتد اوله على مواقع أخرى EE
(лг or лг? or 27r or 2712)
Final Examinations
(4)5—-:-8[- (0 or 1 or 3 or 6)
(5) The additive inverse of (— 5)? is (25 or 5 or -5 or -25)
(6)27+(—3)?2= (-9 or 24 or 3 or 81)
( 7 ) The measure of the angle for the sector of third of a circle is
(90° or 120? or 180? or 270")
ü Answer the following :
(1) The circumference of a circle is 88 cm. Calculate its area. (Consider m= 2 )
(2) Find the solution set of the inequality : 2 X + 1 < 7 where X EZ
(3) In the cartesian coordinates plane ; locate each of the following points
A(1 +1) ,В (351) and C (3 ,3) » then find the image of A ABC by
translation (X — 2 لاو + 2)
(4) The following table shows the percentage of egg production in three farms ›
a merchant collected these eggs to distribute them on the grocery stores :
First Second Third
3596 4096
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى EE
Final Examinations
يهل
Answer the following questions :
[1] Choose the correct answer :
(1) #х+2 3 then x= (-1 or 1 or 5 or ~5)
(2) ZEN U c (2* ог 27 or {0} or Ø)
(3) If the lateral area of a cube is 36 cm? و then its total area = 7 cm?
(144 or 81 or 54 or 96)
(4)-8 -- 2 (€ or € or C or ¢)
(5)I-51* (2 or zero or 7 or 12)
(6) B 22 ree (3 or -1 or -3 or 1)
(7 ) If S is the sample space of a random experiment د then P (S) = зз
(Ø or zero or =1 or 1)
Complete each of the following :
(1)At throwing a fair die once ; then the probability of appearing an even prime
number = ...............
(2)1,4,7,10, ............... (in the same pattern)
(3)A cuboid its lateral area 120 cm? and the perimeter of its base 20 cm. ; then
its height = ст.
(4) fX (-4 ›1)апаҮ (-4 ›—3) , then the length of XY = — و
(5) The measure of the angle of the sector whose area represents $ the
surface area of the circle = d
(7) The image of the point (2 د 4) by the translation (x — 1 و y*1)is
(8) The equation 2 x? + 2 x= 1 is of the ~ == degree.
Choose the correct answer :
(1)An integer between — 1 ,2 is
(-2 or 3 or zero or -3)
(2) The set of counting numbers ·- (€ or € or C or ¢)
(0 or 1 or -1 or 2)
(4)I-111 (> or > or = or 5)
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى [сте
Final Examinations
(5) The number that satisfies the inequality x < — 2 is
(-3 or -2 or -1 or 0)
(5* or 2^ or 10? or 10*)
(7) {1}? , (zer0)?} ~ (€ or € or C or ¢)
a Answer the following :
( 1) Find the solution set of the equation : 2 x 3 = — 9 where X EZ
(6)52х22-..........
(2)A cuboid box with a square base of side length 6 cm. and its height is 10 cm.
Calculate its lateral surface area and its total surface area.
( 3) Use the distributive proberty to find the result of : 32 x 117 – 32 x 17
(4) The following table shows the degrees of a classroom in maths test in
one month :
°“ Assessment ` Excellent
umber of pupils 9
Represent these data by a pie chart.
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى Tues]
Final Examinations
ec
Answer the following questions :
[1] Choose the correct answer :
(1) The image of point (3 , — 2) by translation (4 > 2)is
(05.0) or (-750) or (-154) or (1 ,7))
(2) The measure of the angle for the circular sector of a quarter of the
Circle = sk (30° or 45° or 60° or 90°)
(3) Which of the following can be probability of an event ?
(12 or 17 ог 5° or 101%)
(4) The number which satisfies the inequality x — 2 > 3 is
(3 or 4 or 5 or 6)
(5)A class of 50 pupils. If the probability of success for those pupils at the end
year exam is 0.9 , then the expected number for the pupils who will success
equals E. (50 or 45 or 25 or 9)
Каса (zero or 5 or 1 or 50)
(7) ELLA (€ or # or C or ¢)
Complete each of the following :
(1)1£X (-3 52) >Y (-3 ›— 4) » then the length of XY = --
(3) (4 x 3 + 3) (7 x 3) = =
(4) The surface area of the circle of diameter 20 cm. = --........... X cm?
(5) In the opposite figure :
The percentage of the shaded circular By
sector equals
(6) ( 1)2 — 1 = ee yy
(7)25 521517513, ..... (in the same pattern)
(8)If2y=8 ,theny+3=---
هذا العمل خاص بموقع ذاكرولى التعلیمی ولا يسمح بتداوله على مواقع أخرى [сте
Final Examinations
(B choose the correct answer :
(1)1-31+131=
(2) #х+1=2 ,ћепх= --
(3)35+32=....
(4) NZ =
(zero or 1 or -6 or 6)
where XEN (3 or 1 or -1 or -3)
(37 or 310 or 33 or 32)
(Z or Z* or N or Ø)
(5) The number of integers between — 1 and 3 is ===
(-2 or -1 or 3 or -3)
(6) {zero} = (€ or € or C or t)
(7) The equation : 2 X — 1 = 15 is of the
(first or second or third or fourth)
a Answer the following :
(1)A box without a lid ; in the form of a cuboid its length is 16 cm.
» its width is 7 cm. and its height is 19 cm.
Calculate each of its lateral area and its total area.
(2) In the experiment of forming а 2-digit number from the digits (3 » 5}
Write the sample space , then find the probability of each of the following :
[a] The envent A is the units digit equals the tens digit.
[b] The event B is the tens digit is an odd digit.
[c] The event C is the units digit is an even digit.
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى Tues]
Final Examinations
(3) In the corrdinates plane د find the image of the line segment AB where
A (2 »3) B (- 2 » 0) by translation (X + 3 +y - 2)
551
(4) The following table shows the percentage of the production of a factory of
house electrical sets :
The kind of set Heater | Oven | Mixer
The percentage 15% 40% | 25%
Represent these data by circular sectors.
aeo
Answer the following questions :
C] Choose the correct answer :
(1)92. caf
(2) If zero E 15 و x-2]) then x = ~-~
(3) C 18 = =
(4) The circumference of the circle = -
(> or « or = or 2)
(zero or -5 or 2 or -2)
(-2 or 1 or О or 2)
(лг or 712 or 2xr or 27:12)
(5) The multiplicative neutral element in Z is .......
(0 or 1 or 2 or -2)
(6) The probability of getting a tail when throwing a coin once is
(0 ог} or 1 or 1)
(7 ) A circle is of diameter length 10 cm. د then its area = 2
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى EE
(50 or 100 or 78.5 or 25)
Final Examinations
B Complete each of the following :
is the set of all possible outcomes for a random experiment.
(2) 02) x )- 1(2 «85 —
(3) 4 > 4 > 5 > zs 7 (in the same pattern)
(4) dne measure of the central angle of the circular sector whose area represents
H from the surface area of the circle = ------------°
(5)If X + 2 =|—- 4| then the solution set = ----------
(6)If 2y26,theny-5---
(7)-4[s* - 1] = =-
(8) The solution set of the inequality x + 1 < 5 , where X EN is <.
Choose the correct answer :
(1) The number that satisfies the inequality X > — 4 is ===
(—5 or -6 or -4 or -3)
(2) The image of the point (4 »— 2) by translation (X + 2 لاو — 1) is ==-
((2:-1) or (65-3) or (25-2) or (25-3))
(3) C 100) =... (—100 or 100 or zero or 1)
(4) 1-41-14 | = rrr (zero or 1 or 8 or -8(
)5( |] ع معطاء 2 >1 +ع where XEN (3 or 1 or -1 or -3)
( 6 ) A cuboid with a square base ; its lateral area is 224 cm? ; its height is 14 cm.
„then the side length of its base iS ............... ст. (14 or 4 or 2 or 3)
(32 — E 7 (€ or Ё or C or dc)
a Answer the following :
(1) Find the area of the opposite figure : C UN
2)
(consider x = 22
(2) Find the solution set of the inequality : 2- X> 3 , where X EZ
ED
EE هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى
Final Examinations
:43 х 44 + 43 x 56
( 3 ) Use the distributive property to find the result of
ite sports in one
(4 ) The following table shows the percentage of four favorite sp of
a youth center :
(The favorite sports Lr m | Swimming
20% | 15% was 8р
| The percentage of players | 40%
Complete the table » then represent these data by circular sectors.
Answer the following questions :
[1] Choose the correct answer :
(1)If3xz-9,Xx€Z;thenx*12-- = (-3 or -2 or -1 or 4)
(2) The lateral area of the cube = area of one face x ...
(6 or 5 or 4 or 3)
(3)IfX (2 » 1) and Y (3 › 1) د then the length of XY = -
(0 or 1 or 3 or 5)
(4) If Ø is the empty set د then P (Ø) = - (zero or 5 or 1 or 2)
(5)C3)xI-SI»-
(15 or —15 or 8 or -8)
(6)97 + 95 =.............
(9772 or 92 or geo ор 935)
(7) The next number in the pattern: 2 ,3,5,8 » 13 is -...
(18 or 19 or 20 or 21)
( Complete each of the following :
(1) The measure of the angle of the sector Whose area represents
surface area of the circle = -*
(ә)
22 هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى
3
Ж the
Final Examinations
(2) If the probability of success of a pupil is 2 , then the probability of his
failure is s
(3 ) The solution set of the inequality х + 1 < 5 و X EN ÎS зз
(4) C 1E -12 teis
(5) The height of a cuboid whose total surface area is 400 cm? and its base is
in the shape of a square of side length = 10 cm. equals ---------- om.
(6) 85 = 5 + (8 x 1) *(Bx i.)
(7) lf x=|-12|,y=-3 ,ћепх+у=
(8) The greatest negative integer is
Choose the correct answer :
(1) The image of the point )- 3 , 4) by translation (x » y — 4) is
(C350) or (-7.4) or (3:8) or (- 1:4)
(2)A circle of diameter length 8 cm. ; then its area = ~
(4 or 8 or 16 or 64)
(3 ) The number that satisfies the inequality : x-223is
(3 or 4 or 5 or 6)
(4) lIfa<b,then—3a (< or > or = or 5)
(5)Z2NN= (Z or Z or {0} or N)
(6)-1-61*6* zt (€ or # or C or Ф)
(7) The equation : x? + 1 = 10 is of the
(first or second or third or fourth)
[4] Answer the following :
(1) Find the solution set 0f:2X—-82-26 , where XEN
(2) х 25
(2) Find the value of : 552)
С)
EE هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى
Final Examinations
(3 ) A box contains 4 white balls » 6 red balls and 5 blue balls ; all the balls are
identical د а ball is chosen randomly » find the probability that the chosen Бай.
[a] White. [b] Not red.
(4) The following table shows the percentage of the number of students in one
classroom according to their favorite activities :
[ Activity Sports | Reading | Music | Computer
Percentage 10% 15% 35% 40%
Represent these data by a pie chart.
(в)
Tues] هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى
Www: zakrooly comi
Educational Dir
hid FI-Aeherg L
Answer the following questions :
[1] Choose the correct answer :
(1) C 19 + (199 = ------------- (71 or zero or 1 or 2)
(2)Z2-Ns с (Z* or (0) or Z- or 0)
( 3) The height of the cuboid whose lateral area is 160 cm? and the dimensions
of its base are 3 cm. and 7 cm. equals - cm.
(6 or 8 or 10 or 16)
(4) The image of the point A (— 4 » 3) by translation (— 1 »— 4) is
(05 ,-7) or 5 .- or em or (-3>-1))
(5)lfa6{2,-5,-3} 0 {5 -و 2:-3( ,then a= ~
(2 or -3 or -5 or 5)
( 6) The probability of impossible event = === (0 or 1 or 0.5 or 12)
Choose the correct answer :
(1) (1-91+3) +2 (Є or € or C or c)
( 2) Acube the perimeter of its base is 36 cm. د then its lateral area = mom?
(9 or 324 or 36 or 486)
(3) The number which satisfies the inequality : X > — 2 is «7
(1 or -4 or -3 or -2)
(4) The measure of the angle of the sector which represents i the circle
equals (30° or 45° or 90° or 60°)
(5) 6 1/94 +E qe E (0 or 2 or - 1 or 1)
)6( 32+ 32 + 32 = (28 or 49 or 33 or 29)
E com Complete the он
(1)Z=NU-
(2) Ifx+3=|-7|, then x=-
( 3) The edge length of the cube whose total area is 600 cm?
( 4 ) The set of solution of the inequality : – 2 < X 5 zero іп Z is
ر و ырына нын nod imis ڪي ڪڪ و
and its height is 5 cm. equals -----------
Final Examinations
(6) A fair die is thrown once ; then the probability of appearing the number 5
equals -..- oe
(7) A circle of diameter length 14 cm. د then its area = ~------------ cm? (x = 22
(8)#а=3 و b=-2,then3a b= --------------
4
11
[а] [a] Find the result of : Pr
[b] Find in N the set of solution of the inequality : 3 Xx —2« 7
[c] A circle of radius length 10 cm. is divided into 8 equal circular sectors.
Find the area of one circular sector. (consider 7 = 3.14)
a [a] In a Cartesian coordinates plane ; locate
the points A (0 +4) > В(2,1) > С(-2,1), |
then find the image of A ABC by
translation (0 »— 2)
[b] The following table shows the percentage of the production of a factory of
house electrical sets :
[ Тһе kind of set | Washing machine | Heater | Oven | Mixture
[ The percentage 30% | 15% | 40% | 15%
Represent these data by circular sectors.
№
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى [ЖЕЕ]
Final Examinations
Cairo Governorate
Answer the following questions :
Choose the correct answer :
(1) The set of non-negative integers is (C or Z or {0} or N)
( 2) The equation : 26 + х5 = 100 is of the - + degree.
(11 or 5 or 6" or 15)
(3) If Ø is the empty set د then P (2) = (1 or 2 or 0 or 0.5)
(4) The area of the circle whose radius length is 2 7t cm. is cm?
(4x or 272 or 1256 or 473)
( 5) The integer which satisfies the inequality : y > — 3 is Ms
(7-2 or -8 or О or 1)
(6)If3x-2-9,then-5X2-- (15 or 9 or -15 or -|- 15|)
B Choose the correct answer :
(7) The image of the point e = р " translation two units in the positive
direction of the y-axis is -
(6452) or )2,-2( or (6›—2) or (4;0))
KD CO NUI е апела po. »4 cm. and 0.6 dm.
is - (72cm? or 8.4dm? or 84dm? or 84cm?)
ore (3)? (< or = or > or >)
(10) 2*5 - (Z or N or 0 or {})
(11) Half the T.S.A. of a cube whose sum of its edge lengths is 36 cm.
is cm? (108 or 27 or 54 or 18)
(12) A box contains 14 balls د 5 red د 3 green and the rest are yellow د then ie
probability of selecting a non-red ball is - 3 or E or 8 оғ + )
Complete :
( 1) The ratio between the T.S.A. and L.S.A. of the cube is -
(2) IFA (2 +9) و B(-4:9) » then the length of АВ = -- length units.
(3) The probability of appearing an odd prime number when rolling a die once
Сут Атаат гна WOES وه ASZOTT
®
ر هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى Tease]
(= 3.14)
Final Examinations
)5( )- 3 x75)*(-75- RENE
(6) The S.S. of the inequality 3 + 4 X» —9inZi:
( 7 ) The volume of a cube whose L.S.A. is 144 cm? is ............... cm?
(8 ( The measure of the central angle which represents $ of the circle iS ...............
Answer the following :
(1) Find the S.S of the equation : 2 X 3 = — 9 in Zand in N
(2) Use the distributive property to find the result : 25 х 9 + 25 - 25 х 9
(3) Find the area of the shaded part
»if the radius length = 7 cm. (x= 2)
(4) Notice the opposite pie chart د then complete the following : _
[a] The percentage of the tennis players
[b] The measure of the angle of the sector which
represents the football players is =~
(5) In the coordinate plane
» draw the figure ABCD ; where :
A(351) » В(1,3) » © )3 : 5( and D (5 :3(
+ then draw its image by translation (X — 4 ; y — 4)
What is the area of the image of the figure ?
[стз هذا العمل خاص بموقع ذاكرولى التعليمى ولا بسمح بتداوله على مواقع أخرى A
Final Examinations
Answer the following questions :
[ Choose the correct answer :
(e+ ii (0 or 1 or 2 or -1)
(2)5x522—— (252 or 25? or 5? or 5°)
(3) fx-5-77 و XEN then X= "=. (2 or 12 or – 12 or 35)
(4) The image of the point (4 » 5) by translation (0 »— 4) is ier
((4 9) or (5,1) or (4,1) or (4»—1))
(5) When peeing: a dice once ; then the probability of getting a number less
than 1 =~ (e o OF dell
(6 ) The set of odd numbers f the set of even numbers =
(0 or N or Z or 2)
(7 ) A circle ; its circumference is 44 cm. » then the length of its radius
(22 or 11 or 7 or 14)
(€ or Gor ¢ or C)
(9)/f2x26 »then 4 x (3 or 6 or 12 or 16)
(10) If x 2 > 2 then x €- (N or Ø or Z* or Z^)
(11) A box contains 10 cards numbered from 1 to 10 » one card is selected at
random د then the probability of getting a number divisible by 5 = ~
(i or t ог
(12) In the opposite figure : B A
The distance between the two points =o 2 3 0 3
AandB- its. (2 or -2 or 1
B Complete :
(1)4х32+32-7х3=
(2) #х+3=|-6|,ћепх= -~
(3) The sum of the measures of the angles of the sectors about the centre of
the circle =
(4) The equation : did + 3 = 8 , then the equation is of degree.
(5) А box contains 15 balls all of them are symmetric د 5 white balls » 4 blue
balls-and:the:rest.are-red:balis-; one ballis-drawn:from-the:box-at.random-s-
then the probability that the drawn ball is red = =-
ر هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى Tease]
Final Examinations
(6) The image of the point (— 1 » 2) by translation of 3 units in the positive
direction of the x-axis is -..
(7 ) The lateral area of a cuboid with a square base its length is 10 cm. and its
height is 9 cm. =
(8) In the opposite figure :
ABCD is a rectangle › its length is 12 cm. >
its width is 7 cm. A circle is drawn to touch
the sides AD and BC ; then the area of
the shaded part = ·--- го
Answer the following :
(1) Find the result of :
(2) Find the solution set of the inequality : 2 X + 9 « 1 in Z and represent it on
the number line.
( 3 ) A container water tank in the form of a cube › its inner edge length is 1.5 m.
It is wanted to paint it to prevent the rust. The cost price of one square metre
is L.E. 15 , calculate the cost of painting.
(4) On the coordinate plane :
Locate the points A (3 , — 2) ; B (1 » 1)
and C (3 » 1) » then:
[a] Find the length of BC
[b] Draw the image of A ABC by translation
(х+2 ,у +3)
(уе) ايتداتي/تيرم ؟ ^ / (Worksheets & Examinations) sw رياضيت указ |
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى [стз
Final Examinations
( 5) The following table shows the percentage of the favourite sport for your
class students :
The favourite sport | Football | Basketball | Volleyball | Swimming
The percentage | 45% 10% 25% | 20%
Represent these data by using the circular sectors.
Fast Felurational Zone
(4) Alexandria Governorate pte
Answer the following questions :
п Choose the correct answer from those between brackets :
(10) or 2 or Z or 27)
(Є or € or C or ¢)
(3)Ifx€(2,5.-3)(-5.-2.-3])
„then x (-5 or -3 or -2 or 2)
(4) (9? (> or < or = or otherwise)
(5)C7)- (C1-50n (> or > or or otherwise )
( 6) The solution set of the equation : X— 2 = 3inZ is ..
(5 or 1 or т or {3})
( 7 ) The number which satisfies the inequality : X + 4 > 2 is .. ^
(-1 or -2 or -3 or -4)
(8) A cube of edge length 6 cm. ; then its lateral area = ~-
(216 or 180 or 144 or 108)
(9) The image of the point (- p ) by translation (X — 3 » y + 4)
is )- 5 »—3) ((-8,515) or (-2;7) or (-8>7) or (-25-7))
(10) The lateral area of the cube = Area of one face x
(2 or 4 or 6 or height)
(11) The sum of measures of the angles of the sectors about the centre of the
circle =- e -100*—or 150° or 180° or 360") —
(12) If Ø is empty set » then P (©) = ~ eae (0 or 2 or 1 or 0.5)
(50) X
Tease] ر هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى
Final Examinations
а Complete each of the following :
)1(|-5 | +|7 |= ст
)2( 5 x )-3 + 7( = 5 x )- 3( + 5 x
(3) The S.S. of the inequality : X + 4 > 7 in N İS -..............
(4) In the opposite coordinate plane :
) 5 ( In the opposite coordinate plane :
The length of AC = - ^ units.
(6) If the lateral area of a cube is 100 ст?
»then its total area = ~
( 7 ) The perimeter of the base of a cuboid is 10 cm.
» its height is 4 cm. د then its lateral area = =-
(8) When tossing a die once ; then probability of getting a number 5 =
Answer the following :
(1) Arrange the following numbers in an ascending order :
—9 517 ;|-9|.— 15 and 16
(2) Find the result in the simplest form by using the basic laws of
Pepeatecimuliinijcation 52: 9.
(5)
(3 ( Acircle › its diameter length is 7 cm. , calculate its surface area
22
where n = <>
(4 ) In the coordinate plane :
ABCD is a rectangle where
А (4 51) В (4 *3 C (153) and D (1 1)
»find its image by translation (X — 5 » y + 3)
Final Examinations
(5) The following table shows the number of students participating in the school
activities :
[— The activity Cultural | Spots | Social Ats |
| The percentage 5% 45% 15% 35% |
Represent these data by circular sectors.
ocala Language School
(5) El-Kalyoubia Governorate Ахсым Zone
Answer the following questions :
Е Choose the correct answer :
(1){-3,-4 (C or € or Z or €)
(2)(-71? x23 (25 or 8 or -8 or -25)
(3) #2 х= 10 ,thenx+2=-- (7 or 3 or 5 or 6)
(4) The equation : X? + 3 = 4 is of ... degree.
(1# or 3'd or 260 or 4%)
(5) The image of the point (3 ›— 2) by translation (— 3 » 2) is
((0:0) or (3,0) or (250) or (6>4))
(6) The sum y the measures of the accumulative angles at the centre of
a circle is ~ (90° or 360° or 180° or 70°)
( 7) When throwing a fair die once › the probability of appearing number less
than 4 = ere ( 5 or 4 ог 2 ог t )
(8) The lateral area of a cube whose side length is 3 cm. = ===- cm?
(27 or 48 or 36 or 54)
(9) The number which satisfies the inequality : Xx — 2 > З is
(3 or 5 or 4 or 6)
(10) 28 x 24 = 5 (22 or 212 or 21 or 24)
( Complete the following :
(1) 12 x ---- -72
(2)37+37
(3) A circle › its diameter length is 14 cm. ; then its area =
(4)NUZ--
)2(
ر هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى Tease]
Final Examinations
(5) The solution set of the equation: 3 X 2 = 8 in N is
(6) The solution set of the inequality : X + 5 s 7 where X EZ
(7) A cuboid whose length is 9 cm. » width is 7 cm. and its height is 10 cm. »
then its lateral area = and its total area
(8) The greatest negative іпіеде!
B Answer the following :
(1) A box contains 5 white balls د 9 red balls and 4 black balls. If а ball is
selected randomly э »then calculate the probability that the selected balls is :
[a] White = ---- [b] Black or red = —
[c] Yellow = ---------- [d] Not black =
(2) Acircle M is drawn inside a square of side length 14 cm. and
touches its sides. Calculate the area of the shaded
part. ) = 3.14)
(3) qiue in an злата oer (2 2)3 C5 € s E (5)?
(4) In a Cartesian coordinate plane locate
the points A (4 +3) » B (4:1) > C(151)
and D (1 »3) » then find :
[a] Its image by translation (X — 2 » y — 3)
[b] Area of the figure and its perimeter.
The area = vv » the perimeter = --
[c] Name of the figure. (--------.-..-.. )
kia Governorate
Answer the following questions :
[1] Choose the correct answer :
(1) 1(8 + ) 18 د e (zero or 1 or —1 or 2)
(2) If the radius length of a circle is 10 cm. ; then its surface area = -------------- ст2
(Given that : zt = 3.14) (3.14 or 31.4 or 314 or 3140)
б)
се هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى
Final Examinations
(e or ¢ or С or £)
(zero or -1 or -2 or -3(
(5) The image of the point (— 3 » 4) by translation (0 »— 4) is ( ‚е, +)
(3-0) or (-7,4) or (-3,8) or (-1>4))
(6)Z-Z-= (2 or N or Z* or 10})
( 7 ) The measure of the angle for the circular sector of half of a circle is
(90° or 120° or 180° or 360°)
(8) The equation : x + 2 = 10 is of the === degree.
(first OF second Or third OF fourth)
(9) rr a die is rolled once ; then the probability of getting a number 5
(1 or H or 1
(10) If the edge length of a cube is 6 cm. د then its total area = --
(24 or 36 or 144 or 216)
(11) )- 5) x | - 4 |= (20 or -20 or 9 or -9)
(12) (3 + )3(4 = | (6P or (8) or (3j! or (3?)
B Complete each of the e ماس E
(13) Z = Z7 U c
(14) The lateral surface area of a cuboid = -
(15) In the opposite figure :
The A d of the shaded circular
sector = s %
(16) The probability of the impossible event equus
(17) If x + 6 = 2 » where X EZ » then X = ee
(18) The sum of measures of angles accumulative around the centre of the circle
(19) 222 23 UE
(20) " circumference of the circle = ==
( E Answer the following :
(21) Find the solution set of the equation
ر هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى Tues]
Finat Examinations
(22) Use the properties of addition in Z to find the result of :
— 17 + 19 + 17 (state the property used in each step).
(23) A cuboid with a square shaped base of side length 7 cm. and its height is
10 cm. » calculate its lateral surface area.
(24) Find the solution set of the inequality : x + 4 > 7 ; where X EN
(25) The following table shows the favorite sport in youth centre :
Basketball
Represent these data by circular sector.
Math
Answer the following questions :
Choose the correct answer from those between brackets :
(1) 2-2- = (2* or N or al or 2)
(2) The number which satisfies the inequality : x > — 2 is -- 0
)-1 or -2 or -3 or -4(
(3) The surface area of a circle = zt x === (г or r2 or 2r or 2r?)
(4) When tossing a die once ; then the probability of getting a number 5 =
(zero or & or $ or 1)
(5) (2198 + ) 19 = (zero or -1 or 1 or 2)
(6)/f2x--6,thenx€--——- (N or о or z* or Z`)
Gs)
[ЖЕЕ] هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى Й
Finat Examinations
(7 )IfA(-2 , 1) and B (3 › 1) » then the length AB = === length units.
(0 or 1 or 3 or 5)
(8) If Dis the empty set ; then P(Z) = === (zero or 0.5 or 1 or 2)
(9)(-5)xI4I (20 or -20 or 9 or -9)
(10) If a < b , then: — 3 a —3b (< or > or = or &)
(11) The image of the point (- 3 » by translation (X > y — 4) is - a
(C350) or (-7,4) or هت or ta ,4))
(12) The lateral surface area of the cube = area of one face x
(6 or 5 or 4 or 3)
(El complete :
(1) The probability of apperance a head when tossing a coin once = =
(2)A circle of diameter length 8 cm. ; then its area = T cm?
(3) The lateral area of the cuboid = perimeter of the base x - 85
(4) The equation : 4 x ? — x = 29 is of
( 5)A circular sector represents i of a circle د then the measure of its central
angle =
( 6 ) Ifthe area of one face of a cube equal 9 cm? ; then its total are:
( 7 ) The solution set of the inequality : - 2 > X < zero in Zi
( 8 ) The perimeter of one face of a cube is 12 cm. د then its total area
Answer the following :
(1)A cuboid-shaped box with a square base its length is 10 cm. and its height is
7 cm. Calculate the lateral area.
(2) Find the solution set of the equation: 2 X + 923 .و XEZ
(3) In the opposite figure :
ABCD is a rectangle where its length = 8 cm.
and its width = 7 cm.
Calculate the area of shaded part.
ر هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى Toss]
Finat Examinations
(4) Use the properties of addition in Z to find :
116 + 190 + (– 116)
(5) The following table shows the number of students participating in the school
activities :
The activity Cultural | Sports | Social Arts
The percentage 596 | 45% | 15% 35 96
Represent these data by circular sectors.
Answer the following questions :
@ Choose the correct answer :
(1) A fair die is thrown once د then the probability of appearing the number 6
(0 ا M
(2) The solution set of the equation : 3 X = — 6 in Nis .. e
({-3} or үз} ог {2} ог ©)
(3) #х+522 ,ћепх2> re (3 ог -3 ог 7 or –4)
(4) The integer that lies between — 4 and - 1 is
(7-2 or -5 or 3 or -4)
(5)C5?x(2? (109 or 10 or 10? or 10?)
(6) If Ais an event in a sample space S ›Р (A) = 1 » thenAis = == event.
( „айна or possible or sure)
(7 ) The multiplicative identity element іп Z is - si
(-1 or 1 or О or 2)
(10) or Ø or Z or zero)
(9) The surface area of the circle = ===-
de or Xr? or 27r or 27r?)
(10) The additive inverse of (— 5)? is =- (25 or 5 or -5 or -25)
(11) 27 + )- 3(2 = = )-9 or 24 or 3 or 81)
(^: ¢) одасы 7 / (Worksheets & Examinations) ow رياضيت yasil (57)
ر هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى EE
Final Examinations
(12) The measure of the angle for the sector of third of a circle is
(90° or 120° or 180° or 270°)
B Complete each the followingi:
) 1 ( 2 ZT SN eenen
(2) 14 + 213 + (C 14) = ee
(3) The sum of edge lengths of a cube is 84 cm. ; then its lateral area
equals == ст?
(4) The result of : 23 х (— 1)? + 8 = =
(5) If X + 6 =2 > where X€Z , then х=
(6) (4х3 +3) – (7 х3) = ~
(7) #х=|-3| و y=-2,then2xy
) 8 ( |] - 5 X= 35 » where X EZ > then X
Answer the following :
(1 n The circumference of a circle is 88 cm. Calculate its area. (consider AE 2)
(2) Find the solution set of the inequality : 2 X + 1 < 7 where xez*
(3) ما the Cartesian coordinates plane
» locate each of the following points
A(151) > B(3»1)and C (3,3)
» then find the image of A ABC
by translation (X — 2 ,y + 2)
( 4) The following table shows percentage of egg production in three farms ›
a merchant collected these eggs to distribute them on the grocery stores :
[ The farm First | Second | Third
| The percentage ofthe production | 25% | 35% 40 %
Represent these data by using the circular sectors.
|
ر هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى EE
Final Examinations
El-Dakahlia Governorate
Answer the following questions :
L Choose the correct answer :
(1)1-981- ГАШ (€ or € or C or $)
(2) The image of the point (--------------- » =-=) by translation (X — 3 ناو + 4)
is (- 5 »- 3) (C8; D or (-25-7) or (-2:7) or (2>7))
(3) The equation : x? + х= 5 is of - degree.
(fourth or third or second or first)
(4) The probability of the impossible event = ~- (1 or $ or 1 or 0)
(5) C6 -—12 (> or = or < or <)
( 6 ( A circle › its diameter length is 20 cm. د then its area = -= cm2 (m= 3.14)
(31.4 or 314 or 23.14 or 43.14)
(7)2-(€3)°= . (5 or 3 or 1 or 2)
(8) The sum of edge lengths of a cube is 24 cm. د then T.S.A. =
(16 or 36 or 4 or 24)
(9) £X(3 58) + Y (354) د then the length of XY = n length units.
(4 or 6 or 12 or 5)
(10) If (S) is a sample of a random experiment د then P (S) = ممم
(0 or 1 or i or i
(11) f 3 y = 9 و then y + 5 = re (11 or 32 or 8 or 14)
(12) The additive inverse of (— 3)? is -............. (9 or 3 or -3 or -9)
Complete :
( 1) Two things must be known for the translation to happen
( 2) The probability of the sure event = =-=
(3) C 1)190 + ) 1у103 = .
( 4) If a cuboid shaped box with a square base its length is 9 cm. and its height
is 10 cm. د then the L.S.A. = em?
(5)C69xC2)7
(6) The measure of the angle for the sector of third of a circle = ==
| ر هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى Tease]
Final Examinations
( 7 ) A cube ; its volume is 1000 cm? ; then its lateral area = ~-
(8)2x32«32-4x3--
Answer the following :
(1) Find the solution set of: 3 X —7 55 › where X EZ
(3) х(-3)*
(з?
(2) Find the value of :
(3) In the coordinate plane :
Locate each of the following points
A(2 +3) +B (4 +3) and C (4 :5(
> then find :
[a] The length of BC = |епдїһ units.
[b] The image of A ABC by translation (0 » — 2)
(4 ) Find the lateral area and total area of a cuboid without lid » its length is 16 cm.
» its width is 9 cm. and its height is 5 cm
( 5) The following table shows the percentages of production of a factory for
three kinds of electric water heaters :
The kind | 4% | 28 | за
Percentage | 25 % | 35% | 40%
Represent data by the circular sectors.
Final Examinations
Directorate of
Directing Mal
Ismailia Governorate
Answer the following questions :
E Choose the correct answer :
(1)0z*nz-- (Ø or 1 or -1 or 2)
) 2( #2 X =0 then X= 2 (2 or 3 or 5 or 0)
(3) The greatest negative integer is a: (2 or 1 or 0 or —1)
(4) If X + 6 = 5 , then the solution set in Nis ~ s
СЕЙ or {1} or Ø or {0})
(5) |] ع 2 +ع« | - 5 |, ћепх= =- $ (3 or -3 or 7 or 4)
(6) The solution set of the inequality : X > 0 іп Z is ^
(Z or Z* or Z- or N)
) 7 ) The image of the point (3 » 0) by translation of magnitude З units in
the negative direction of X-axis is ~
((3›3) or (050) or (3 »—3) or (0.—3))
(8)Ifx>y,thenx+z (> or < or = ors)
(9) The probability of the impossible event = (Ø or 1 ого or -1)
(10) The surface area of the circle = 7t x . (r or 2r or r2 or r3)
(11) If a fair die is rolled once » then the babii of getting an even
ubora Se (0 or 1 or
(12) If the total area of the cube = 54 cm? , then the area of one face = :
(4 or 5 or 8 or 9)
B Complete :
(1)Z*-Z-^2N--—
(2) The sum of edge lengths of a cube = 120 cm. و
then the lateral area = ----
(3)у— 4 > 215 an inequality of .............
(4) The area of the circle whose diameter length is 14 cm. = =- cm?
(5) On the number line : A
The length of АВ
length units.
(6) #1х1=3 then x
(7 ) If one of the families spends its salary as the following 40 96 for food › 20 96
for house rent › 30 % for expenses د then saves the remainder is
1
s 0 4)
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى EEE
Final Examinations
(8 ) A cuboid of leng бст. » width 4 cm. and height 5 cm. د then its lateral
area = ст2
Answer the following :
(- 2)5 x35
1) Find the value of :—3,——
(1) M 33 x (- 2
(2) Calculate the area of the opposite figure.
(Consider w= 22
(3) The perimeter of the base of a cube is 28 cm.
Calculate its lateral area and total area.
(4) Find the solution set of the following equation » where X EZ: X+5=4
(5)A box contains 25 balls » 6 balls are yellow ; 7 balls are red and the
remainder is black د if a ball is drawn randomly.
Find the probability that the drawn ball is :
[a] Black = ~- . [b] Not red -
Suez Governorate
( Answer the following questions :
)1[ Choose the correct answer :
(1) When tossing a die once د »then the е probsbalty of getting a Бире! on the
upper fat ri-orioreo)-
(2) 10] e (c org er cione) ‹
ر هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى EEE
Final Examinations
(3) The equation : x? + 3 = 8 is of -— degree.
(first or second or third or fourth)
(4)1-51-- (< or = or > or otherwise)
(5) 15+ ) 8 )-1 or zero or 1 or 2)
(6 ) The sum of the measures of the accumulative angles at a point =
(90 or 180 or 270 or 360)
(7)If2x2-6 ,ћепхєЄ............... (N or Ø or Z or 27)
(8) 5 (< or = or > or otherwise)
(9) The total area of the cube = Area of one face x =-
(2 or 4 or 6 or 8)
(10) On the number line : А B
= 4-32-1012 3 4 5 6
(8 or 7 or 5 or -2)
(11) 5х (-4) = mm (-20 or 20 or 9 or -1)
(12) The image of the point (— 3 » 4) by translation (X » y — 4) is
(C350) or (-7>4) or (-358) or (-154))
І complete :
(A) ZANE cm
(2) The circumference of the circle =
(3) 232 =
t 2 » x€Z,thenx----
( 5 ) The lateral area of the cuboid = perimeter of the base x
(6) A cube of edge length 10 cm. ; then its lateral area = ·--------------
тоо е = (length + width) x 2
(8) A box contains 5 white balls د 3 blue balls and 8 red balls all of them are
symmetric. One ball is drawn from the box at random. Then the probability
that the drawn ball is red =
Answer the following :
(1) Use the properties of addition in Z to find the result оѓ:
(— 7) + 19 + 17 (state the property used in each step)
ر هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى Toss]
Final Examinations
(2) Find the solution set of the following inequality in Z: X 2 < 3
(3 )A circle › its radius length is 7 cm. د calculate its surface area. (where л
{4 ) А cuboid shaped box with a square base. Its length is 10 cm. »
its height is 7 cm. Calculate the lateral area.
(5) The following table shows the percentages of the production of
a factory of house electrical sets :
The kind of set | Washing machine | Heater | Oven | Mixer |
Thepercentage| 25% | 15% | 40% | 20% [
Represent these data using circular sectors.
(12) Port Said Governorate m ui Renter | mj
Answer the following questions :
[ сһоозе the correct answer :
(1) The surface area of a circle = JU x ==- (r or r? or 2r or 3.14)
(2) If-2 X= 6 then x €. (N or Ø or Z* or Z^)
(3) The number which satisfies the inequality : X — 2 > 3 is
(-1 or -2 or 6 or 4)
(4)C 08 + ) 1(9 = ee (zero or —1 or 1 or 2)
(5)15- 11] (# or E or c or £)
(6)25х22=- (27 or 2* or 23 or 1)
( 7) When tossing a die once the ‘probability of getting a number onthe upper
face more than 6 is ==- (Ø or zero or 1 or 2)
©
هذا العمل خاص بموقع ذاكرولى التعليمى ولا بسمح بتداوله على مواقع أخرى TEE]
Final Examinations
(8)1-31= s (3 or -3 or -|3| or 3-3)
(9 ) The total area of a cube = area of one face х =~ i
(4 or 5 or 6 or 8)
(10) The probability of the impossible event = ~ (@ or zero or 1 or 2)
(11) The image of the point (2 › 3) by translation (X + 1 لاو + 2) is
((854) or (355) or (4 ,3) or (5 ,3))
(12) f xe6-22 و x€Z,thenx---- (4 or |-4| or —4 or |4|)
Complete :
(1) 3+[-3|= m
(2) The perimeter of the base of a cuboid is 10 cm. , its height is 4 cm. » then
its lateral area = مس
(3) The probability of the sure event —
(4) The sum of the measures of the angles of the sectors about the centre of
обе m «eet
(5) The circumference of the circle = ٠ “хл
(6) A cube of total area 150 cm? , then the length of its edge is
(7) Z* U {O} = те
(8) #3 X= 9 s then X = ere
Answer the following :
(1) Find the result of : (4 x 32 + 32 — 7 x 3)
(2) In the coordinate plane locate the points
A(253) › B(453) » © )4 ,7) then find:
[a] The length of BC =
[b] The image of А ABC di translation (0 »— 4)
(5:0) теуш! \ / (Worksheets & Examinations) ow رياضيات wall
| هذا العمل خاص بموقع ذاكرولى التعليمى a اوله على مواقع أخری [стз
Finat Examinations
(3) Find the solution set of the inequality : X — 2 > 3 where X €z
»then represent it on the number line.
(4)A cuboid shaped box with a square base its length side is 10 cm. and its
height is 4 cm. » calculate the lateral area.
( 5) The following table shows the percentage of the production of a factory of
house electric sets » represent it by circular sectors :
The kind of set | Washing machine| Heater | Oven | Mixer |
The percentage 30% 15 96 40% 15%
3) Damietta Governorate 0 биза mieria
Of ficial Language Ceh
Answer the following questions :
(Ei choose the correct answer :
OZAN.. (Z or Z* or {0} or N)
( 2) The equation : X3 + 4 = 5 is of the ~~ degree.
(first or second or third or fourth )
(3)A circle » its radius length is 4 cm. » then its area = ===
(4 or 8 or 12 or 16)
(4) The image of the point (— 3 » 5) by translation (X + 1 sy — 2) is
(C453) or (-253) or (-2,-3) or (2,3))
( 5) If a fair die is tossed once د then the probability of getting an odd
number = - (0 or 1 or Т ог })
(6)|-41-I41= . (zero or 1 or 8 or — 8)
(-7.) АЙ the following numbers satisfy the inequality: x > — 3 except --
(zero or 4 or 1 or 2)
©
هذا العمل خاص بموقع ذاكرولى التعليمى ولا بسمح بتداوله على مواقع أخرى TEE]
Finat Examinations
(8) The sum of edge lengths: of a cube is 96 cm. »
then its lateral area = ·--- = om? (8 or 64 or 256 or 384)
(9)A circular sector represents 3 of a circle د then the measure of its central
angle = ·-------------- (90 or 120 or 180 or 270)
(10) If 3 x 2-9 ,then XE... . (N or Z* or Ø or Z^)
(11) )- 1 + (— 1(9 + ) 1(2 = . (zero or -1 or 1 or 2)
(12) The solution set of the inequality : 2 < X < 3 where X EN is --
( (zero) or {2} or {3} or {2,3})
Complete each of the following :
(27 × 2(8 _
210 e
(14) If 22-3 =|—7 | then x c
(15) fX (-3 2) » Ү(—3›— 4) > then the length of XY = - - units.
(16) The height of a cuboid whose lateral area is 160 cm? and dimensions of its
base are 7 cm. and 3 cm. =" cm.
(17) A box contains 5 white balls د 3 blue balls and 8 red balls , all of them are
symmetric » one ball is drawn from the box at random د then the probability
that the drawn ball is red = ===
(18) The multiplicative identity element in Z is .
(19) The image of the point (— 1 » 2) by translation of magnitude of 3 units in the
positive direction of y-axis is
(20) The surface area of the circle =
(13)
Answer the following :
(21) Find the solution set of the inequality : 3 x- 224 , where X EZ
(22) Use the properties of addition in Z to find :
115 + 390 + (— 115) (write the used property).
[стз هذا العمل خاص بموقع ذاكرولى التعليمى ولا بسمح بتداوله على مواقع أخرى A
Final Examinations
(23) A cube of edge length 12 cm. Find the total area.
(24) A circle ; its diameter length is 14 cm. Calculate its area where (x =
(25) The following table shows the rate of the score of 200 students in one
school of Cairo governorate :
[ Rate [Exceten| Good | Pass | Weak
[Percentage| 15% | 50% | 25% | 10%
Represent these data by circular sectors.
heikh Governorate
Answer the following questions :
Choose the correct answer :
(1)1(X—223 then X 2 зз (-5 or -1 or 1 or 5)
(2) The lateral area БЕ cuboid of length 3 cm. » width 2 cm.
and height 4 cm. - em2 (20 or 24 or 40 or 52)
{га cien “г (< ог > or = or s)
(4)3-|-3|= р (0 or 1 or 3 or 6)
(5) The image of the point A (3 » 4) by translation (1 »— 1) is sr
((353) or (2,3) or (4,3) or (4 ,5))
(6)Z*NZ-= (Ø or Z or x or {0})
(7) 1)104 + (– 1)103 = — (zero or —1 or 1 or 2)
(8) A cube of edge length 6 cm. د then its total area cm?
(36 or 72 or 144 or 216)
(9°) adie ls thrown once ; then the probabitty of appearance of
the number 5 = ----- 5 or 1 or 0.5 or 1)
@)
C6 6
ЙЗ هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى Tess]
Final Examinations
(10) The area of the circle = хл (г or 2r or r? огг+2)
(11) The measure of the central angle which represents 1 ofthe circle = ver
(90* or 36° or 45° or 40°)
(12) If S is a sample space of a random experiment ; then P (S) = =
(0 or 2 or 1 Poe)
B Complete the following :
(13) if x+ 5=3 ,X EZ then x=--~
(14) The perimeter of the base of the cuboid is 10 cm. › its height is 4 cm.
»then its lateral area = 2
(15) The equation : х2-3 = 615 of the .......-....... degree.
(16) 32 + 23 =. E".
(17) If the perimeter of base of a cube is 20 cm. ; then its total area is
(18) A circle of radius length 7 cm. د then its area
(19) IF X (- 3 و 2( » Y (- 3 » 4) » then the length of XY = ~
(20) The probability of the impossible event is ...............
B Answer the following :
(21) Find the solution set of the inequality : 2 X + 1 > 5 , where X EN
(22) Find the result of :
(23) If the sum of edge lengths of a cube = 36 cm. Find :
[а] Its lateral area. [b] Its total area.
(24) A circle of radius length 7 cm. is divided into 8 equal circular sectors.
Find the area of each circular sector. ( = 22)
TEE] هذا العمل خاص بموقع ذاكرولى التعليمى ولا بسمح بتداوله على مواقع أخرى A
Final Examinations
25) The followin
(25) anarad 5 table shows the percentage of the number of students who
р: in а School activities represent the data by a pie chart :
| The activity Music | Sport An |
[The percentage | 25», 40 % 35% |
O El-Fayoum Governorate Educational Dirocforafo
Maths Inspector
Answer the following questions :
@ Choose the correct answer from those between brackets :
(1)NUZ^--— (z* or Z- or Z or N)
(2) All the following numbers satisfy the inequality : X > — 3 except
(0 or -2 or - 1 or -4)
(3) 1)11 + (— 1)10 = -- 25 (zero or -1 or 1 or 2)
(4) 23 ,x€e€Z,tenxz--—--- (5 or 7 or -7 or 6)
2
(> or = or > or 5)
(5)I-71* 351-77 +31
(6) The additive inverse of c 3p is ~ (3 or-3 or 1 or -1)
(7)Ifx-4 » y=—3>then the negative number of the following is ~
(x+y or x-y or xy or ух)
(8) The image of the point (4 » — 3) by translation (X — 3 » y + 3) is -
(C75—-8) or (150) or (0,1) or P »6))
(9) The probability of appearing a head when tossing a coin once =
(zero or 2 or 1 or
(10) If the probability of success of a student in mathematics is 75 96 د
then the probability of his failure = -=== (25 or 0.35 or 1 or
(11) The ratio between the lateral surface area and the total surface area of
(223 ө ат сакы ел)
00 cm? and area of one base 20 cm? د
-em?(40 or 60 or 80 or 140)
acube-
(12) The total surface area of a cuboi
then its lateral surface area =
[2] Complete each of the following :
(13) The degree of the equation : x3+3×2+ ×+ 4=
(04:4) оида / (Worksheets & Examinations) =ш ريه يات palad]
TEE] هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى D
= 11 іѕ ~ - degree
Fina! Examinations
(14) The solution set of the inequality : X S0 in N = —7777
(15) The solution set of the equation: x +6 = 5 іп =
(16) if the perimeter of one face of a cube is 20 cm. د
then its total surface area = ---- ст“
(17) In the coordinates plane if the point A (— 2 » 4) and ће point B (5 , 4)
»then length of AB = units.
(18) A cuboid its lateral area is 120 cm? and the length is 8 cm. » width is 4 cm.
ə then its height = ------ em,
Ci
(19) Teamference of the circle Be
5 the probability of any event s ----
B Answer the following :
(— 5(5 x (—
(21) Find the result or 8267
(24) A box in the shape of a cuboid , its length is 10 om. , its width is 5 cm. and
its height is 8 cm. › find its lateral Surface area and its total surface area.
Football | Basketball | Handball |
40% 35%
Represent these data by circular sectors,
25% |
Ре
هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى EE
Final Examinations
( (16] El-Menia Governorate кишен EON
мл.
Answer the following questions :
Choose the correct answer
(1)ifx-2=3 ,then x= . (-5 or -1 or 1
(2)A cube of edge length 6 cm. ; then its total area =
(36 or 72 or 144 or 216)
(3) When tossing a die once; then probability of getting a number divisible by 5
equals ...... (0 or t or 5 or 1)
(4) The equation : x 2 + 3 = 4 is of the ce degree.
(first or second or third or fourth)
(5) Тһе smallest natural number iS ............... (0 or 1 or 2 or 3)
(6) The number which satisfies the inequality : X » — 2 is E
(-1 or -4 or -3 or -2)
(7 )A circle, its radius length is 4 cm. » then its area = zt cm?
(8 or 16 or 64 or 2r)
(8 ) The additive identity in N = + (zero or 1 or -1 or 2)
(9 ( The total area of a cube is 324 cm? , then the area of face = = En
(54cm? or 81cm? or 54cm. or 81cm.)
(10) (— 1)104 + (— 1(103 =. (zero or -1 or 1 or 2)
(11) The probability of occurrence of the impossible event =
(Ø or zero or 1 or i!
(12) If - 3 х < 30 و then X ^ (— 10) (> or > or = or <)
[E complete each of the following :
(1) Measure of angle of the circular sector in which its area represents ع 1 from
the area of the circle = -
(2 )IfX )-3 5 2( لاو )- 354) » then length of XY s ~~ length units.
(3)2*-2- =
(4) The lateral area of a cuboid of length 3 cm. د width 2 cm. and height 4 cm.
ر هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى Tus]
Final Examinations
(6 ) The image of the point (2 ›— 1) by translation (X — 1 »y + 3)
is the point (-............. an)
)7( Ifx+3=|-7|,thenx=~
(8) Ifx=|-12|,y=—3.>thenx+y=
Answer the following :
(1) Find the solution set of the inequality : 3 X — 5 < 7 where X €Z*;
then represent the solution set on the number line.
(2)A cuboid ; its length is 6 cm. » its width is 4 cm. and its height is 8 cm. Find :
[a] Its lateral area. [b] Its total area.
(3) Find the result of : 222
(4) A box contains 8 white balls, 7 red balls; all balls are identical; if one ball is
drawn randomly; find the probability that this ball is :
[а] Red = сс [b] White = --
[c] Blue = <... [d] Red or white = =
(5) The following table shows the percentage of eggs production in three farms
during one month :
[ The farm | Fist [Second| Third |
| The percentage of production | 25% | 50% | 25% ]
Represent these data by circular sectors.
(QV : ¢) а ^ / (Worksheets & Examinations) ыш رياضيت yalsell (з)
| هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى |82
Final Examinations
s
Answer the following questions :
Choose the correct answer :
is the smallest positive integer. (-1 ого or 1 or —10)
(2) 2+ NE 9 ({0} or Ø or Z or zero)
( 3 ) The probability of getting on the upper face of a die a number which is
more than 6 when tossing it once is <... — (Ø or zero or 4 or 4)
(4) The surface area of the circle whose diameter length is 20 cm.
= == em? (л = 3.14 (314 ог 0.314 ог 3.14 ог 628)
(5)(—1)8+(—1)9=- А (zero or -1 or 1 or 2)
( 6 ( The probability of the impossible event = ·- (0 or 1 or 2 or 3)
(7) A circle » its circumference is 88 cm. » then its radius length = -= om. (д = 22)
(28 or 24 or 44 or 14)
(8) The equation : 4 x? - x = 29 is of ............... degree.
(fourth or third or second or first)
(9) The smallest non-negative integer is ·-- се (1 or 0 or -1 or 2)
(10) A circle ; its radius length is 7 cm. د then its area = ·-------------- cm? (x = 22)
(145 or 154 or 22 or 7)
(11) The image of the point (— 4 د 3) by translation (— 1 ; — 4) is ٠
((55-7) or (-5,-1) or (-753) or (-3,-1))
(121-9143 —————Z (€ or € or C or gt)
Complete each of the following :
( 1) The lateral surface area of a cuboid of length З cm. ; width 2 cm. and
height 4 cm. = ~ 2
2!xc2P _
(2)©
(3)2=
(4"y tf the perimeter of base of à cübe 15 20 cim: > then its lateral’ area
(5) IFA (2 »4) و 8 (2 »— 1) » then the length of AB is - units.
:
ر هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى Tease]
210
(6) In the opposite figure :
The percentage of the shaded
circular sector = ----- ~ %
(7 ) The sum of the measures of the accumulative angles at
the centre of the circle = ===-
(8) The image of the point (2 » 4) by translation (X — 1 » y + 1) is
Answer the following :
(1) Find the solution set of the equation : 2 X — 3 = — 9 , where X EZ
(2) A cuboid box with a square base of side length 6 cm. and its height is 10 cm.
Calculate its lateral surface area and its total surface area.
(3) Find the solution set of the inequality : 3 X 2 < 4 ; where X EZ
(4) In the opposite figure :
ABCD is a rectangle where its length = 10 cm.
and its width = 7 cm. و $e the area
of the shaded part. (7t — 22
(5) The following table shows the rate of the score of 200 students in one
school of Cairo governorate :
| Каќе Excellent | Good Pass Weak |
|. Percentage 15% 50 96 25 96 | 10% |
Represent these data by a pie chart.
Answer the following questions :
)1[ Complete :
(1) If the lateral area of a cube is 36 cm? , then its total area = —
)2( )- 18 + 61) =
(3) The distance between the location of a number and the location of zero on
the number line is called
(4) The additive inverse of zero is
(5) The image of the point (3 د 5) by translation (x + 2 » y — 1) is
(6 ) The probability of the impossible event
(7) IfA(-2 51) B(3 5 1) > then AB = - units.
(8 ) A cube of edge length 6 cm. د then its lateral area = ~
Choose the correct answer :
(1) If S is a sample space of a random experiment; then P (S) = =
(zero or 2 ог 1 or 0.8)
(2)-1-54|9 r (-54 or 54 or 9 or 1)
(3) The geeen Me uS integer is ~ b (0 or 1 or -1 or -2)
(4 or -3 or -5 or 0)
(5) Type of central angle of a circle is straight angle » then it represents зз
from surface area of the circle.
(quarter or half or third or whole one)
(> or < or = or otherwise)
(7 ( When tossing a die once; then probability of getting a number 5 = 000000
(zero or 4 or 2 or 1)
(8) If the perimeter of base of a cube is 24 cm. ; then its total area = cm?
(144 or 36 or 54 or 216)
(9) The equation x? — x = 29 is of the - degree.
(first or second or third or fourth)
(10) If 2 X 2—6 » then X Gen (N or Ø or Z ог Z)
(11) 15 + 3)] x x)= (22 or -22 or 88 or — 88)
(12) Z* --- “N (Є or € or C ort)
1 3
ر هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى Tease]
Final Examinations
Answer the following :
(1) Acircle ;its circumference is 44 cm. Calculate its surface area. (7t = 22 or 3.14)
(2)A cuboid ; its length is 6 cm. ; its width is 4 cm. and its height is 8 cm.
Find its lateral area and its total area.
C3Pxca*
3) Find th: It of :
(3) Find the resul me
(4) Find the solution set of the inequality : 3 X 2 > 4 where X EZ , then
represent it on the number line.
(5) The following table shows the percentage of the production of a factory of
house electrical sets :
Marks Washing machine | Heater Oven Mixer |
sw | 15% | 40% | 15% |
Represent these data by circular sectors.
(19) Aswan Governorate i
Aswan Official Langu:
Answer the following questions :
Choose the correct answer from those given :
(1) The greatest negative integer is ---..- ` (0 or 1 or —1 or 2)
(2) The total area of cube = ^^" x area of one face
(6 or 2 or 4 or 3)
)©
А هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى TEE
Final Examinations
(3)1-6|*16|2 n (12 or – 12 or 1 or 0)
(4) The image of the point ( ..-.....-..-..- متم ةع ةسام و ) by translation (x — 3 » y + 4)
is(-55-3) (78515) or (-25-7) or (8,7) or (-257))
(5)(-8)x1= (-7 or وه or 8 or -8)
(6 ) The probability of the impossible event = ·-------------
(0 or 1 or -1 or d)
(7 ) The solution set of the equation : X + 2 = 7 » where X EZ is зз
(-5 or 9 or 5 or -9)
(8) (= 36) + (= 4( = ee (-9 or 9 or -6 or 4)
(9) 7-]-3 |E ss (21 or -10 or 10 or 4)
(10) The previous integer of (— 9) is (-10 or 8 or -8 or 10)
(11) If 2 is the empty set then P (O) = == (zero or i or 1 or 2)
(12) The image of the point (1 »— 3) by translation ( - -> -)
is (1,0) ((150) or (0,0) or (3:0) or (0:3))
Complete the following :
(1)IfX*622, X€Z then X
(2) 3)? =
(3) The lateral area of a cube its edge length 5 ст. equals
( 4) The image of the point (3 » 5) by translation (X + 2 لاو — 1) is -- e
(5) The total area of the cuboid = + the sum of the areas of the two bases
(6 ) When tossing a die once; the probability of getting a number divisible by З
equals -
(7)Z-N=
(8 ) In the opposite coordinate plane :
АВ = e Units.
Tess] هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى Й
Final Examinations
Answer the following :
( (1) Use the properties of addition operation in Z to find the result of
the following : 37 + 25 + 63 +75
ircumference 88 cm. Calculate its surface area. (Jt
(3) Find the solution set of the inequality : Xx — 2 2 3 و X EZ , then represent it
on the number line.
(4) A cuboid shaped box with a square base its side length is 9 cm. and the
height is 20 cm. Caiculate the lateral area and total area.
( 5) The following table shows the percentages of the production of house
electrical sets :
[ The kind of set | Washing machine | Heater Oven Mixer
Тһе percentage | 30% | 15% |
Represent these data by circular sectors.
PI) South Sinai Governorate FT frelon! Zom
(Answer the following questions :
[1] Choose the correct answer :
(1) 3 ————6 (> or < or = or <)
(2) #2 X= — 6 then X E» (N or Z* ог Z- or {-4})
A
[её] ر هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتداوله على مواقع أخرى
Final Examinations
(3) The image of the point (3 » 5) by translation (X + 2 » y — 1) is
((556) or (5,4) or (1,4) or (156))
(4) When tossing a die once; then the probability of getting
a number 5 =- (zero or 4 or È оғ1)
(5)|-65] (Є or © or С or ¢)
(6) The number which satisfies the inequality : x > — 2 is ----
(-1 or -2 or -3 or -4)
(7 ) The circumference of the circle = =: x TU
(г or 2r or i? or r+2)
(8)Z'nZ-s eii (Z or N or @ or (0))
(9) If X is less than — 5 »then the symbolic expression is
(x>-5 or x«-5 or x25 or xs-5)
(10) The number of faces of the cube = =-= faces.
(6 or 8 or 12 or 4)
(11) The sum of the measures of the accumulative angles at the centre of the
circle 4 (180° or 360° or 270° or 90°)
(12) Ifx-2=1 معطاء x=. (1 or -1 or 3 or 2)
Ш-ро Дын | me
( 1 ) A cube of edge length 6 cm. د then its total area =
(2) If the base area of a cube = 49 cm? then its lateral area
(3) IFX + 5 >2 ,then X» ~=-
(4) The probability of the impossible event = ~~
(5) The image of the point A (1 » 4) by translation (X — 2 » y + 1)
is the point A (
(6) The equation : 3 X2 — 6 = 14 is of the ............ degree.
(7 ) If the perimeter of the base of a cuboid is 10 cm. and its height is 4 cm. »
then its lateral area = - em?
(8)1£X (7-352) Y (C3 ,— 4) » then the length of XY s length unils.
TEE] هذا العمل خاص بموقع ذاكرولى التعليمى ولا بسمح بتداوله على مواقع أخرى A
Final Examinations
(2) Find the solution set of the equation : 2 X + 9 = 3 , where X GZ
(3)Acircle » its diameter length is 14 cm. د calculate its surface area.
The surface area = зз
(4) In a Cartesian coordinate plane ; locate
the points А (2 » 3) »B (4 53) »C (4 » 7)
and join them; then find the length of BC
The farm
centage of the production | 25%
Represent these data by using the circular sectors.
(0:0) Y ابتدائى/تييم ^ / (Worksheets & Examinations) su رياضيت ука]
TEE] هذا العمل خاص بموقع ذاكرولى التعليمى ولا بسمح بتداوله على مواقع أخرى A
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apes cre, өц jo esse su |ә]
qun stet. = e~ BES =
Pie = apup eq jo eere eu o] ام
jun HO ZE = 9500 - grs =
uod peunofoo et jo eave еш.
ue 96007 =
de) х عزن = erro eu jo ver әш.
ЕЗ4
ссх д} = әбиле oq jo ею әш [s]
юу = 3202 - wi =
ed peinofoo ец jo eave sur.
ange
d£) x FITS = apne ou jo eee eu
quo Wi «9» G} =блра ou jo oor ou] [a]
quo ov zi = SBL- 96°02 =
‘wed рвлпосә eu jo көзе eur
دوو ص
(AS) * УГЕ = apap powe өц jo воле өц
صب 96002 = |
8) еу = oro ofi ou jo толе өш [e] fj
рону»
LL- 019 = Wed ратоюо ә 30 ел ou].
amus уф =
Spuonues оц jo ооло UL
жерех Ё +
Spaoxuoe ou jo ue) snipes OUL
wa ph =(0} + 01)- وم =
өрдэшәв әд jo uibus) pou оцу.
жөо 9L ve = билш )
= اوو ص
G'86 — pg = Med penojoo әд jo ease ou].
qose.
de f = app ajo cave оцу
дт = £ x zi = umes ou jo wave ац EY
jn وج = 01-98) = ed рагоро өш jo tare eu.
اهو صلا = Ad) x fy = ошо ou jo сөл eu.
дабу = vix = arende oq jo vore әш.
дв {шг=®+ 00+ £n =
А
ө хо+ sex gr deh
зод oy! lu]
ge 91296) = STOL + 929:96 =
| 901х001 + (وجفار х ё = eese ou, I5]
ро És-» Kes
rex f gen f x T = om ous И
ميج 9901 *0/ +98Е=
01x 2+ fo) dg = vtm ous Ll
фета = 09 + STB! =
акаже) » fy x ў = кае ou [p]
زوه روؤصع «do
gx px Ê +e) * fx ع moa ous fot
posees
% + 92008 = JOA auo jo waue ou.
مسر 629060 =
ASTI) хере = vei ю әзе sadan jo eave ou].
‘up رجو = sz» < «беј spa ou (
BESTEL = a+ روز = юри әй jo pare ө]
quo vil = D» & = weep sijo sam sui (YY
zao
$ + EOS = 0008 euo jo eae eu
E= W ошо өш jo seve өш (ff)
go dz =+ 562+ £c عاو
gos وار 52+
ано ع حي
uo wo Lx «= suero eu]
лу = ) x dy «opio eq о веш oul
шоген
eese eut [1]
bua ميعسعيس ax 2 = Bue snper ou @
3008 Up әш; jo £iomsuy
ров UDH oi jo saoMeuy иш
هذا العمل خاص بموقع ذاكرولى التعليمى ولا
بند
اوله على مواقع أخرى росто
jun 012 = (62 x 2) + 09) = зл oy ou
240 92 = $x 8 = veue eseq au [o]
م = و صن + 02-9 seq = о бие өш [a]
040 091 8x 02 = Bove reser өш [e] f
= <<
1995 ра
зад эш jp opel er
NE = rey eu.
e 0E =z (e+ 1) = ood өц ю apod ец (I
cd
3 ممه = ير segou
180 زوة = ووة + 022 =
(ех |8) +022 = wave mor әш.
0109991 = زوه + 9801 =
(EX ovz) + 9805 = ease eyo} out.
gi? ObZ = 02 x 2} = Bane seq оцу
HP 9901 = Д x vg = геле reee ou]
wo وم - 26x 2=
@ «0 «tnn ono سس eu ы @
‘89x09 ج00 =
330 + 901 = soxoq Jo equo eu
j 80! = %06 x DL = ве респ ous
quvso-
E0% EO x9 = ovo ouo pave гүл әш EEJ
Abs pXIXLSgkzxzo
'qno әш oup jo eave caso
оц + өдпә веб өф jo Pe [610 SU] =
وميد ө рю во түл eut GE
S707 31 و ع хосу = биро jo оо ou].
U S'EL = GZT x 9 = төл! мүл өц
00902-91 1 = 008 euo окат ous
ДИЮ جوع = (уа « Z) + OEE = Beve ло; ay,
j0 p9 = و عاو = вале eseq ou] [q]
uo 026 = 01 0
jD BOZ = Z X YZ + رون = Bowe jojo eu]
صني PE = px 9 = Bowe эчер оц}
jO Q9) = @ x 07 = ear» поје өш
woes Z x (p + 0) seq ou oed a
gun OVE = OL» ye = Bove ome әш وي
UP Ob = »اج Oz = eere pesoyej aL
"WP 02 = Oj «2 =
(82 + 9'2) x Z = эзе eq jo مس هوعد e [о]
019 806 = 901 + 008 =
бах 6) + 009 = вала јао әш.
yt ون = @ × g = eae oseq эш.
019 009 = GZ х وج = Bare јеле au]
физ =9 < 9x9 = awon ou 7
лш و =\дбоә opo өц >
9x 8 = эбрә x apa ~
qui ®% = эое) эшо jo ваше eu -
quem = v» 96 = eere раа эщ
e08; oup сол әш | _ دج روح و د عو سر
мю? «v» và = Бале pope зи Lo]
"uo g= bue epo өш >>
9X в = 06pe x өбрә . [а]
qu^ عون + و = وو = ee ouo jo ге оц [е] f
jlo دعوو ICD
qun 961 = Рх برع = воле reste UL
ubere 251900 jo eave su).
"uo Le p + gp = бон spe ош GB
quo جو = 9» gc = cove [eor әш [9]
UD pph = y% 96 = vare елде ميات
j عو =9 x 9 = ee) auo jo төл oa [a]
"unge у + pz ufuer eBpe ous [el (Û
quio 0001 = O1 х OL» OF = өшлюл ou [>]
دج 008 = 9 x 00у = өле joi oup [9]
quo 00h = у> 001 = weve eseye, ou [o]
рю 00V = 01 X زو = e ouo о wale өц!
чоо = г\ + رجن = Чоо ope аш (A)
quo FOZ = 9 6p = ete rept ou [а]
gin زهو = x бр = weve زعرصم ou [v]
دلق Gr = £ x Le 9984 suc jo eere eu.
"uo بج ع + їй = uus әбрә ou @
2029001 шон ie) әҹот Ы)
ї= Ê 390 _ SPO _
610-2 = oe = eae ae our ©
UP هوم = 9% p9 = тэт o өш
090 992 = j x 49 = гале [uote] eu]
WO وج = у> - وده el مر юшде эщ [a]
32 وم = ха = som ouo р vore әш fj
uo 6و = 9x 9} = Bae ies еш [o]
iub وز = vx SL = Бэ шюр] әш
guo رو =p ў = 08] 9v0 jo eeu өч fa]
изу = وو مرج = yiwa دمقه өш [s] EY
ди => x g = еме юлды әш.
جع + و = و ناخ = 608) euo jo ear eu fj
lS ps يدوت 6 = еме io әш
Bewe (ej ош = ج جو gei = Dx صا
gun جهو ssi = 208) ouo 0 care cu 7
дю BE = § y9 = кәл erc UL
uo »دم = جوع vo = воле ею UL
юз = в ха «oo ouo jo eave eu (f
مسج 912 = 9x DE = кэе екл out.
gla yy) = px Өс = eee роја SUL
وو ميج = 9% 9» 998) ono vere ош (E
йа = 8 х pph = wove poi 041
дї = yx vi د e | ош.
qu P =Z} * Zh = 608) auo jo esie ou] [o]
gp yat = 9» yo = eae до UL
E 900 = зеш yo UL
quo 006 = t x 922 = wave [oie OUL
quo Set = 91 x 91 = ao) auo jo еме әш Tel I
DITE
х) - fO LL = esie роторо әш
مس 96212
HE) уу = эрле yews oq jo etit ou].
UM
49)» وترم = эрир ебе әш jo eere өш (9
Чу رونو = Ней pepeus оц jo вол еш.
quo е = хэ = af uza! oy o вше аш.
gio tat = (وار x уг = эрэр sio sere ou (fg
КТЕ
12) х & =чәрлб ош jo vorm әш >
g0 = + 96 008 a0 р عند ouu ©
quo yar = эх جم = зәл гул ош f |
w = z + zb = uibus] enpe аш * [а]
m— -ps p= ee 1
3008 шэн аш yo saamsuy
pog ири гр, jo огнот Ww
هذا العمل خاص بموقع ذاكرولى التعليمى ولا a
بند
اوله على مواقع أخرى расо
diuo 09 = Zx ŞEL + оар = vere epo ац
800087 = 0} X фу = Bare елдщ эш.
“wo gp = (6 +91) Z= ostq ә jo жш eu
guo Sê} = GL x 6 = esum ose eu
wo gh = gos = ردقي +
quszooo=
SZ0* S70 = эң ouo о eem әш.
i 909 = 00 + saz = ease [eon ац
du 00v = 01 ^ бу =o sui jo vary
ا 02 =
SOZ *(O + Op) دج по resi ош Ie ©
“wo 6 = garg = pM‘ os
wo: 8 te
0: s5 8
une : busy : фри
чю уа = OL — bE = рїн + إسقريا og
"шз = ‘up Lube ә
uve «у, = бю + шр + ve! f
кдхај= va
"шо 9 pue Шоу
t pua
ouo zt wo om сиоршөшр eq eunssy (f
39 002 م =
007 9 009 6 = ware aded purewai өц.
aUo 00V 9 = 06 « 08 x و = eure |до өц
qun 009 6 =
08% OZ} = рисдрдеә ay jo sore ous (E
"spunod osz عن =
$20 € + SZZ 22 = 1809 pmo) ou.
“spunod $20 £ =
S09 9 = Bujonoo jo eoud eu]
'epunod 922 /z =
909 x Sb = шево yo soud su, [a]
ооа eae =
97 + 0896 = 5ахо jo Joquinu ou
“sonora 6 =
52900 + ووو = son مر sequinu eu].
WO 91 Д 90 + وهل = зен sequin au]
د ومس
ох وج = әк soo vo au]
d 289 = 005 + 292 = هندع [oy eu]
d 00у = 01 * GZ = oo oup jo Bory
quum
98% روا + SZ) n دع вше erae ننه
emper
5900 + S99 = Som jo سوه ош.
2д#50900= »مجو 5202
өң auo jo eave eu,
01999 د 006 + 99) = more тул ou,
iu 008 = رج X جو = 1004 94) jo сөлү
quse
SZE «GL gz) «po eoe emm ou (EY
TOLL 31 OF x HLL = бше р CO I.
2 2842 = (2X 90 + BBL) = ها =
woos ei jo Hed роит әд مر eere roi эш.
dU 90 = 9'0 x | = wopUm euo jo ease au.
00861 = CZ * 60 юор ә jo Base Su].
gu جو = هز + 06 = Bao лор Op
Qu وه جو x g = Bujeo ou jo ve
quos -gzx px g = eom eee) eu (Og
20009 3-1 = »ده буу = бийдей jo 100 eq]
pueste-(goxzegi)-ar-
оо! eu) jo wed pared ор jo varo jeg oq
dU иго = 1 L9 = Mopu buo jo eere edt.
iugi = 2> 6'0 = юорещ о wave eu,
x y p = еше eme) oui GD £ = جه سو
"YES ЯЛ = 8x 3:69 = биле jo eoo эш
«91s = eure jeg ay 02 = له سن
Oz = $ p = бий әш psa سي
| jugs = z2 (p + S) x z «тше ею ән
шор = (Z +g] x Z - اه وعده рио! GD)
oz و - DL جز = But jo оо эш.
quie 2x9 + 91 = eee таеш
و نار = zx دع eee ese U
gust = Ê bx op eom une өш.
FEL ЗЛ = Ob * ŞEL = 9000 оош.
QUSEL= vts 6Y = eave Imo UL
j جين =9 x = таю әзе UL
868-
ко» (9'1 + 5'1) = z = sore rue эн 9
105 3 1= 9| * r£ = 20001200 941.
qui FES = O} + Кё = eae удо] OUL
إن بج = giz x y = бйно jo гою эц
jura.
g1 X(T?) х2 = rove ممع ou ED
ЖЕЎ 372 »اج S96 = 920 зоо eut
jl SOE = 9€ > و + E = эзше ү эш
ELE 9x 2= vow woe ou |
эз! «a XE = бишей jo 1800 OUL
qug -0L +81 «ene таеш.
إن لال = Fx Sz = cow eseq uL
gam = p% (re gE) x = eee el ou (B
O89 = 00У + ج09 = ээл moi eut.
j 00? = OL x оу = Reve seq eu [a]
диб = SZ x 00 = көзе ree өш.
"woo = (01+ OF *Z=
7 (a) 161
(ul (fel
(tal
эзе et о js ead au fe) ١
дош] дой
quo ون = جوج x + о = ше ie өш.
Quo و0 = 21 x 9 = веш 9680 ou].
ex (EL +x z= tore eene aul زوج صل
"uo g = 255 = родго eq jo fren эц. 3
087
juo YEE = 990 Z + 969 b=
(VZO 1 x Z) + 989 1 = Реле 1640) OUL
qun эш | = ze X Ze = Рал aeq өш.
дю p= Z1 X gt) = wave esane әш.
таз = zex $ = Орч эц
un gz} = p» ze = эке eu юеш oun 9
14 * جو = ploqno әщ jo Dave [eee] PUL
wo چو ©
(є + 8) x دج өвва родго өш jo sejowlied оці
ue oor =
رون عدي = вало eu jo ease |20421 OY)
M
‘01 хо = eqno әш ә soey өшо jo eere aus. (f
E TCU
Quo ون = 6x2 = езе озш эш.
w=- &-
uua әй - улпа = PM эш.
quo oze = 0p x ze = eave prov; ous 63
3008 цәй әй jo siausuy
оов шоу эц; 40 взатсму иш
هذا العمل خاص بموقع ذاكرولى التعليمى ولا à
بند
اوله على موا
قع أخرى Ts]
juosie=
مونو - 04 = ved papeys aq jo ware au].
ЕА
Aged & = ала э مر eave әш.
mm
2 x DL = өбиерләл eq jo вола әш [9]
quo دوعو × $ = إجو әшпол әш (c)
диз OS} = عدو عاو 8 = vore тү әш, (z)
240001 = px و عدو «ваш prp әш (1)
wo 9 = zi +09 = бие әрә ө [е]@
“sun qiva بو = 2% )6 +2) =
QOBY aume: eu; jo sewed au.
"Sun uate = 28+
suun õue) z = gy (z)
жов ору ац jo sasusuy
jn 2ê = x 0> $4 э! = CO pa UL
يور صن = × ege son mem ouu |! 3
qun جوع هوم =
eqno үгшбро aup jo eave о әш, = eqno
өш 30 wed рөшешез eu jo eere вул eu. (EY
OL XZ + G9 = воле о) әш. »اج = مون ماج
jun ع وز 9+ vee =
eqno su مر eot эшо jo wave au f)
ЕТ
goge x(Le lx
о:
wng : fue?
шоу = 02> £ «sumo оцю 4
u*up-—122
(в: а=: 28
(piDy-—( ovi
uo وزو = ei) x ф = eame ous ~
1
2x
vei lP]
{UP مون = 0} * 9 * g = enon sit
guo عرق = * 09 + 957 = эше pci OUL
از 09 =9 хо = 8 9ш
jus pgz = ¢ x وج = ware шер әш.
шоу زوع +) +91 =
(бузц + ирт + Bue г г
sig
= ypa + иби,
eos шон = soc. Й
هذا العمل خاص بموقع ذاكرولى التعليمى ولا
بند
اوله على مواقع أخرى EEE
"هذا العمل خاص بموقع ذاكرولى التعليمى ولا يسمح بتد اوله على مواقع أخرى ]1525
seele) њо: 00181
sv el «co pl
2081 191 o0 1B)
giesno Aq од [a]
asid vole
zu
"no pouse er updo eu
маци поо ра вәшодио әзәш jo OA
eds adus [q]
ameu 5 1529] әд, vods بع вош эш.
far
аи: FE =
uode jo ofun puo ө yo enseou au,
par
ошер jo oue рдо эло suneou ou]
Jy e oc x ¥ =
Smau jo обо [адио aq jo MINEOLA eu].
46-096 x ¥ =
gurino jo аула puso әй о огош eu]
406 = 09€ х 9 = Guruıeyaue
jo eiue едиво өц jo einstou ou.
‘anol 98 =
I+ L+ $+ $ +6 = sinou jo wns ou gj
eros о efus [Mao مر بيه anseou au
= єх =
usua jo еби enues өш jo aunseow ou,
9 » озек Be
soups jo абзе оци ө о sinseeu SuL
ШЕ perm
sous jo ous peguen э jo алгаш ou
з= лг» =
эделу مر ojbue reijues ош до олтезош 341
001 37 80e OL=
‘Anes puooes eu jo Keres ќош eut.
OBO 2371 - Z1. 06 = 894 әш u cones әш.
06 7371 = 006 %0 = juo بن pores eu.
ej
p
(406 + %02 + £09) = 600) = مومه ош @
O8 TT = 0021 х %0} = Gunes ay, fa]
HO) = (ML +
% 09 + % 2) % OOF = вол әш C0] 029
носа щом 24 Jo зоташ
= س
] بمب وم ә уо =v — |
هذا العمل خاص بموقع ذاكرولى التعليمى ولا ي
بند
اوله على مواقع أخرى EE
ee | هه alg
RIR 28 8
зе 20
эе qs Eqs әвпелед _
зои rs ө Gursootp s seq ou, °°
Behe B = vanes
8end puooos ә leu) امهو ع وزيم ou
алоо فرعيو vx eu io بموعوبرية ou]:
парте oce lal 6
sorto)
0z ( lal
ê
OLeogaxi
OL =X = sui jo joquinu.
X = Sho jo soquiny
ia
E
3.9.
eo
20 e
-g-a $.9
Qo.
E]
0= "bx 06 am
مر Ape} e jo eouenue jo yyqeqoid eu, [o]
Ê = $ =-B ои veg оош Bram
$ = 92 = ои ueg sol jam
مر Ape] jo ocuenuo jo Amago әш [в] 9,
== alal
En = Wag
Ё = ppo я aoe sadn au uo
Jequinu jusvedde oy) yeu انمو دجوي ец [ә]
$ = ياج eoey sadin әд vo маши
wovedde әш xeu Awasqoud оцу [a]
estu {erz hsm
Fag mun:
Lene ue бицей مر Ayaqeqod әщ [ч]
А
= Ê аши
awyd ppo ue бидәб مر Amaeqoud eu; [e]
6د [re ic ezeiz ei ai) سه عو
E:
>
гроаб, об juspnys stu zeuy де он әш.
‘SWOPMIS 99 = B+ اج + |9 + اج +
арте де jo зәдшпи әш
$ روه Bp om om
Jo jonpoud eui æu Aurqeqoad ou; [p]
$ =p
oni ә مر uns eu req Ayqeqaud эи [o]
2.8
РЕГЕ
ppo задр suun өш ien Арек! $41 [dl
Н
i-
ppo si Hp sues әш reu Аишдедол au [e]
eztu
dcs ezt »وو eos ez! eei ze ze zz} - ©
scd 9181
0211 œm
2 2 - 001
#{-{-@аш $- 1
ve (s)u* [99*59
$o
өш
чш
[1]
iw
il
“BUS OFZ
09 — 001. = uswom jo sequinu ou] <
ai +
“oul 09 = scs = бәш jo sequin әш.
$. +
Стата
يي „ залови ,
DO u el.
еа 09 = gq s = мед опа مر жашла eu
0
oa
ga теит qun all _
эте enr оладат ац
bp
пед aniq e Buweup jo Ápiqeqord eu; >>
X = 1q pei а Suwesp jo Аалдедол әш > 8
i-9-gam $=
{ors ل ge nee:
{=ош
ع «Wall
:sü
all © دع
ocv 29{ “نزو »ل
75199 02=4+5+8
لق
ft
7982901 +قع 6+ 7
9 ice
$-Fm 5-9
“sieq 02 = ZL + 9 = Sea we jo qun ец,
= (aa لها
„аа
35 ET]
dessen ezi)- s
1008 цон эд jo sionsuy
X008 щом 343 jo sams b |
f هذا العمل خاص بموقع ذاكرولى التعليمى ولا ي
ty
وله على مواقع أخرى EEE
e 3
| 3
E
3
4
S}ƏƏUSYIOM JO 3
ѕләмѕиү opinc) J
>
SA 1
залиш sı ) Ey i
si) $e) o E Wey 3
«ûf = „дә x Ê = sons =
Е торов jo әбив алго әш о глпәвәш ou. EM
ES
D
áo of £o i-fog d
|
(oor سس aun зо sioner МИ
о JOC и оз, د« + Асо dex
'D
i
^1 grep) = S'S OUL >
тахт ваха c+saxz ia
S104 دجامو + 9¢ eei | A
ба * evz [0] olpl
puccos [e] · |
{r-}= езеш
axe
{a-pa وها eu +
M-2X6£" جدورس لماع«
{i=} ونوا eu
ex g-c-=x ما
62 5вәш т b-ag“
у-ахут Brel
(9) «ss эш > gex
бехет —— veuexelol
{s} ='s's uL © Sax
Si-X€" адеб
{s} = вш © Gex
Q-xz7 — beeexz ig
GEH
|
|
әбәр ot) saap pt (e)
sabap ре 00 вевәр yl (H 19]
SU -6*S2-,£*,5-, .£*, sie)
ий 2
Ber‘ ро ‹ 26 [0] 2191 veli
{e+ z} fal 19
(оси {9g p+ es z} tot
С Erol tog
lol {r}! ош
ште
QE
se) тїр] оч 2018
ЖЕ)
SENE
ERE UE]
co: cv: اع ©
CREIO] + ot ipl
seid een
ox
«29 pue
MEN ZEN EET
[DC
X ==) ze-= · 1
2 < ЕСІ]
„а= 822-4)
rege لماج ssi
‘S6
LIT
= etl iss st
ol) sip] leb] в-їй эе
OE
)06 + )06( عبج نر عبج
зазцэнгом JO sams
= 001 + 001 = ج00
= )9+ وها « )90 + 9(
© روبك ب+عوعور
ve
عوعم
=7+169+9
{з
1x £82500 * CL = —001) x g4 I0]
Ld
Lx tg 00} * 8 = +008) va [el 03 |
б=ЕхЕ- (6 +2-)є(0 141
00012 - =
{az~) «0001 = (ex (6) x (szi xs)
беу} = şt x 00 = mx (z= os) 1) 103 @
s-o вш вш 2-0
orf] нш 1-01
[a] бәм samsod e [e! GB
Qu
L =t Lı: ерю әш [e] ع عنم بر
g^ 0* c — sion әш [e] f عمسمو
0012 +0 ع o01
(его) + (ev + 99) = + عوو)
= وو + جو э + (ost)
Saz + gy + (090-) « co fal
ez=6z+0=
درو + رومهة
sr +вё+5-14@
вели expe [p]
Anuepi ennippe [9]
әһдејпшшоо [д] ميدق ميم
Kuedo;d anjaposse [е] 0
obl sip 9-9 هادة 5-1 @
Gu
SRE IRS Ol سوج v8] ol
seez uso} fo)
ti
e EL о1-0-6- +
oll gil sebl ой
2619 ог ebl йс
эн >ы эш 29
иы її 2-b сй omg
si s-ip] 2-1) sta 5-14 4
© iun uo siesusiow
{ sisoussion jo sisnsuy. ١ |
| هذا العمل خاص بموقع ذاكرولى التعليمى ولا ي
بند
اوله على مواقع أخرى EEE
шони = de x) = авец
^uo £=2 + уу = фбїюң snipes oy (2)
wows бн
оро oq jo ooussepusraso әш (4) [el f
UO 92:66 = 9 + PLE = юрәв ouo jo eave ou
)x pve = eae ou E
ЖУ = эше ош fp]
(9) » Pye = әле әц [о]
qui 42°09 = ДУ) х v £ = vore әш fal
quo د 81و (9) vic» vore әш [е]@
Gum
0
Aq O8 Y ю obeus еш э gy V“ os.
G- D -»-*t«z39
SEL
G-zez3gatzzJep
(2584 Ds I) vo teu out [o]
sun 9 = |9-|= [Z-t -|= ові
"sine |е | |1 -z-|- gv le]
Aq DGV V jo обош eq si Оруу, os
€-2-
(t *£-* p-o) osie- (و ojo абеш өш
(rete
(b+ bs زر “)بقع ترح gjo өбеш ous
n=
( «e p-s) уз! (e © spy إسعقه مر oy)
ıı = Anon jo sexe jo aur эщ [s]
‘3600800 زه SHO V [d]
an; jeij} + زع 72-6] - o Pel
siun eenbe zi = px = oam eu].
"wapa
2% L= ZX (e+ у) = Jevoun0d ouy lal
чиш з دع [z+
жипте «9x yx C esum ou [1
"eun osano g =
دع بدو Ẹ «ogvvio esie eui [о]
aw
peifiue-u&u pue
г euejeos s1 ogv Y 18]
191 )- 91 + 91( +62 = 0 + 6€ = هم
E اها (ө! =) (ссх v) = 001
D هذا العمل خاص بموقع ذاكرولى التعليمى ولا ي
‘p= Кдашиќа jo saxe jo oqunu ou [o]
"Sun arenbs gz = g x ç = eoim باه
"spon og = p x وميد دو өц [o]
auenbs =) әйецв eu] [е]
بند
| pez-o مه مد زوج on
ezto
اجهمه هاه
{оч ع[
esx T95
[797v وهاه لدع oul
ss веха
sext sext
ا ا esse
fsx SKE
ic3-z|- xd
i-misxes espere il
jo suamsuv جزل توبات هرد
اوله على مواقع أخرى Toss]
эздацәро 50 HOMEY T
(r*Qg-—tz« na
09.0) و سسنج
(9*9 g-—t« Ha
(p+ 9) у«— (2: ИУ
{шени
1H1* HHL * L1H* HLH‘ LHH“ HHH} = s 8,
+04 = 096 x BÊ = цебиз
Jo iuu enueo et jo eimseau eut.
409 « 000+ S „елер
зо afue [enueo ө jo einseaui eut.
406-095x 8 =
مر еби enues eu уо ensseu еш i
x 01 = sopas билет. 096 = وز
эбе eaueo ei jo emnseeui ed] « مر
x 02 = юрөз eo 096 = 21
Jo абое үедоәэ 9uj jo einseaur ay) +
«HI = 096 x 201 = opos роду
(jue enue ө jo eimseeu eq « مر
x @ = юрез ues 09¢. = 48
«06 = «098 0 = оров avons
مر оис үедїө oup jo exnseou өц.
«ZL = ле x FF = rojos usua
مر әјбие [едизо ayy jo a neeau эц.
«06 = .096 x SOF юров шен
Jo sibus jenuoo oq o aunsoou au] +
804 = .озЕ x ® = ороз one
go абие fenua әң jo amseou eu] « [a]
sz lel © ]
mdndo = gae
og x 20k e end орохо ю ижи ou [a]
496 = .096 x 90. = орз qua
Jo 9bue jequao әш jo cunseow ou] +
406 = р x Bt soos ssed
مر aue елиш e Jo әлетөш әд].
sZt = „008 x FF = oops ممم
مر биз jenuoo ә jo елтоғәш eu] .
+ 096 х у = юрәз مو عدر
مر әбоо оло oq Jo onsoow oq] . [v] fj
gu
"wo 2 = OF + Ori = Nbre ou].
uo 02 =
CX (y + 9) = өзе eu jo seewyed өш [a]
quo OL =Z + 02 = өзе eu jo Bove oul
دعو مين
روج برج = saseq omy oq jo ease әш [e] ©
quo زوز =) x {= eere eu [а]
guooiz-
»جو Z+ 091 = vore рү ош.
و »ود جو صم -eseqou уоевзе эш (c)
"ung = у +02 = apis eseq jo бшщ au (z)
guo 09i = 0x 02 = ease esr әш (011 @
Quo رون = 8x جو = еме امه( UL
guo SZ = y + 00} = 09) ouo jo wave ou [a]
Quo رهن = 9 og = core eiae ou) [o] f
Jo сзаношу رهد اعبات قرع
| هذا العمل خاص بموقع ذاكرولى التعليمى ولا
بتداوله على مواقع أخرى E
suoneuluiex3 еш јо
SJOMSUY әріп
«801 осн =
шау „p jo бос ели вд алзезш eu
06 = 008» Ot =
"wm, gê Jo fos айл at jo ammi OU
xj ار jo fiue 1060 84 jo ansaa au] [a]
ую جو 9% = زوم = vore ею eut.
302001 = rx Sz = vove jese eu].
quo و × و = جو = оор ouo jo vos өш
uag ду -uifiva әбрә әц [e] E
«Sp = 00» $ = юров одо
apuu دمناريمر jo einseow eu] (z)
mosala
$ + $91 = 0008 evo jo wase eu.
ans sax =
apap 244 jo эол әш (1) [a]
وج صن =Z% وج برع Ope эзе امه өш
Do] ED ريه رعسم yx (Z+ c) = vox جن صب
Z} = seque pai jo одит eq >
= aie
aa FT" ё
sere kejo cum su
Sepmaperp шташ = ed ou @
oci £5 £m 29
{ هذا العمل خاص بموقع ذاكرولى التعليمى ولا 2
بتداوله على مواقع أخرى E
rg
dun y9 = qp x у = tare gero ош.
موصي = ү} * э = тәге илет әш (ol ED
(rewil frs) э of HG
Ho wo we wu
2m uo 20 208
i oct) cul
571136015 spaau 1013905 au;
Jo] uoljipulU/DXe 13262
Suoneujuex үш jo siàmsuv.
ولعي و رع
(ses g-—ie* 9g
(4-ez) y (e+ avt
sun مقن + 08001 |,
= - 06و - el صمو
‘ved pepeys ou jo eose әш.
Base әш. مر x = apro (ع 6 BID SEE
бтр jo ease au] [a] = و »ع = 01208
[z-)-sscu*
cay Boxe
-sxs шут
$-s-xz- Sage
Tr ras
ФЕ-=@-0е-=є+з-0-@@
HU
n2
sid
iu
Di
iu
nue
200
هذا العمل خاص بموقع ذاكرولى التعليمى ولا
بتداوله على مواقع أخرى E
pia)
ie- 9) (2)
99+ - sox
natus pano ао epo
abbl = х x FO = ues
Jo бие eauso ад jo unio out
$8 = عون, * DD = meag
مر fue лого әд o ай саш au]
ZL = .096 « Bt = ounou Buysom
سيدا ساديدا ЫШ
(вск es^ Se er} «s (2)
quo 906 =
4* э! + هرو = 69/0 RO L
quo VU «61 * Sp = се шн эш.
Suoneupuex3 [eur J0 sraMsuy
Pm
o bue enuo ay) jo ainseow PL
«08 = .036 * f = poof
jo fue jeauso oi jo emnseow ou]
+84 - 6و0, x $F = рооб Lon
ojus jene eq jo uinseau ot]. مر
9*9* وج Ope = vam imo эщ,
P OF = OF = pz = еше joe өчү
"wow.
عدو 9 = әзе өш jo seemed adl {Z )
St) ots
2l ale) Diz) امه
(ع) بهم
из) EI
9 geen (2) fa
Mab ate) als)
B0) sie) 202) s-
зиоцешшехә (грош jo ззәмзшү
suoneujwery |еш4 jo siamsuy
هذا العمل خاص بموقع ذاكرولى التعليمى ولاب
بتداوله على مواقع أخرى E
vi X 2x 2 = обид oy jo ease аці
اوو صاز =
gl) * fg = ouo aw jo wove ощ (0)
Qz'S8911:H23X UUM
(s-)» vsu izax umm
ms ose x OE uono
Jo aue jenueo au) jo алзвэш au
abs = дз x 901. ورور
عام ا mone) i amo wl
«801 = 098 x 30 = өшчоеш Buysen
Jp sue jenueo ou jo anseouw su, [a]
$09 uon (o
“wo 0 00) viz)
eo ols)
Ир экш
ote) £-i8) (.—
ade) 20)
suonpuiupx3 ,sjootas Jo siamsuy
suojeunaex jeu 10 SIMU
iai
NI
abet = 096 x 004. snndwos
M TCI
7.096 х 00 = osnuw
Pope pe oq ннн
و1 = „ор x Gt = бире,
مز fue rers» әп jo боғе ou,
«9C -.озс « OL = suods
efu fenus» oq ja arisen әш (v) هر
Q+92-=XZ*
paa (2) $19) nis)
sis) tz) oeny
AE
suoneutux3 [eut jo 523 suy.
هذا العمل خا
ص بموقع ذاكرولى التعليمى ولا
بتداوله على مواقع أخرى E
оос 20 = = اج
ague uso p ainseow ми. Len در )1309
Et
8-S-XZ7
36 є-()
2i nee) _ waz)
WEZSNOR
abusa a]
e 3 1 “ديعم دقام
(E) ريه is epo
qni.
Lu el тов ea om md
pue
s.m.
£573
Aper lou si req ou ie Ашшдедол ац [p].
Lau Marne
ptt
ота є هه وهر лац, Киндэ ou [o]
ge
iun sr eq ou ye ночо ош fe] (0 (f)
(9i-)g-— leva
ГДЕ i vt
وو صان = fee)» يك = ee эш 0
d
se (2)
р” Шш 8-*в1—:знере ou (ы
-@ qun opp шо oze (D
tot bez} (9)
{2)0
zin
эз (e) e VB
э(е) seda)
оос) (0* 0) (8)
Lis} 90
ru d
مر (610 eusa оч jo rao ea
ъс ео x BE = reoos
р ue peuso э jo өлеш өш.
azos = „000 « 00 = spots
р ave eausa ә Jo aseo эч
81 =.оз mid
jo бие eave) эч مر ense эч (3)
Suoneunzex3 еш 10 SJamsuv
ov)
ost (9) (ot
(251010) (e) zug
оф)
Jo бое enuan әл jo ainseaw ou
«06 = .09¢ » = jeaKogon
JP обие jequed oy инеш өш.
Р aue euro oy jo einceous ouy (5)
0 зшопешшехэ јешз Jo 54ә®$иу
هذا العمل خاص بموقع ذاكرولى التعليمى ولا ي
بتداوله على مواقع أخرى E
(Gal مصخ БОО
g ы
B sim
يهم 100 s ед аф e لنموع وبري әц [n]
=.
т =
юв 5] وهر әш leu Ampgeqaid out [e]
“SHE д =
Ч + 9) - 92 = siea eq jo jequn өш (s)
{Jess ow »
vent
(у) مقعم
jI YZ = 9 X 1x 2 = еше eon eu
iio OBL = v x Lx 1 = Вале (een ou]
p + 92 = Bua, epo oq] (c) دع "un
әлбүө oet ou] درن + 082 LSE مسج
ајалојшаѕ әш jo Base out (2)
se =6 x> Ex 220 ©
pun مزجي ejGue |елизо jo ainseaw әш.
Л
„921 د .098* Fer =
punt „д jû абе jeziuso jo eunseaut 241
001
409562 = 6 х 91 + 002 = cove ero; رياه
200 052 =S x و0 = eave relate) ayy,
"woos-
ZX (6 + 91) = seq eu jo жиашиәб эш (9)
(e*9o-—(s 2
(eg ولمعت
(° ya 2 vla)
zeli
e)
exem
{tze et h= ss oug
rsx =
sxe nexo
1*85X£ 7. ssi-xe- ug
o-is) о» on (9
zip oie)
i
6-00)
v(6)
net
puooas {e} (2*2) (2)
o
abl = 098 * =
FINE =
Рия) 10 әбие еләео jo ainseaw eu. (81
шејрда p абое enuo jo anseo эш.
suoneupuex3 jeug jo siowsuy
iR
шеро p sue яшә jo amseaui au]
Е 06 = 096 «2 =
ur ave jemueo елогошащ (0)
1-25х27
دار 919 = (rt) x бе eave эш.
Sl) сг
аа) cun
2ч _ eig
ami) aie)
ودع e
©
bs 3
Ht Jo ou» مصاوع jo ainsesw Эш.
D
w= оох She
Jenos jo бше ели? jo өлїзвөш eu
уаз = ose x Bhs
sods jo өбое enuso jo aanseaui eu).
زو = 09e x She
euro jo sue ieaueo jo eanseow ou (s)
06t = 081 + 0= 06! + [0112 « 91] =
ов «(911 «911 = (1 لح + 06) +81100)
uP Sus =ззе-э5=
ved popeus әш jo eue eu,
posses
dst) y тою
в-{=хг@+ =x
юа: = L> × ضيه D
c Aen
"m
wwe wa so
. и «ton
oz-(e) «ez (8) #0
2249) use(s) t
ad 1-6 nto
ıo еби [aun jo eunseeui аці.
TT ÉL suopeupuexg peut Jo ssamsuy
هذا العمل خاص بموقع ذاكرولى التعليمى ولا
بند
اوله على مواقع أخرى EE
Mods jo бие jenuo jo пасош ou
„06 = 09 «Bt =
энш jo обие jenueo jo апчето eu (s2)
1 2 дове euo joven ац
247 @ = ооло ou jo eose ө (2)
vsi teu) ost Ш)
puoous (sy) Ov (r1)
LEN мш st
BON jo бов anuos о eunsesu ou
06 = 090 « St =
sad مايه anit parsons,
ле] = .09¢ x 2 =
буза jo Bue enua jo esnseow ou (8)
وان 09, =p Qe "өле зәл LL
“wo op =
xO} = وموم eu jo snowed ous (p)
19 s WEE rz 5 م te
.5 :مهمه
(7*1*9*s) sss
saxy росах ez-x^(0
AB, M
"
4
0
4
t
H
H
(co سج ونه
(-nge—tc: а
ü-av-—( avid
“suun uiuo, + = o8 [el (2)
d--iz-ve
Etri- 6n v0 ~
s sl
зое
сөс (9) сәг (6)
puvoes (c) >t
Eon
suoneujurex3 тешэ jo slaMsuy
ѕцопецшехэ وإناع| J0 ззамзшу
| هذا العمل خاص بموقع ذاكرولى التعليمى ولا
بتداوله على مواقع أخرى E
Z * (p+ 9) = oseqou jo зјәшџәд eu (7)
doen = pc fy sento т
А
att
رص as раис
Dia a-on
pun 6) = giz (a?
<t) жч)
4-0 к
vit (e)
oz (8)
x 90 = eon 098 = عور
Jo обо jenoo2 jo ainsa эц
x 00 = sed 096 = 08
бив jenuao jo ainseow өш مر
лә: = nec ^ ML = ood
ou] نمع [enum jo ans همقر مر
gc SE iuo وز
Jo бик едиго jo avos out (s)
عزاو دسي = 0-01 >
uad papous әш o east ou]
jue sees
x @ apap эд р вож ещ
эї5иерәз ац jo gare eu] (y) = بو برع
бәш = زج نو نومع
ark
Факт sexe
cesexes razed
aout.
enr Pio. دجوو مج عو برو
5пегшшеяз решу јо s1amsUy
ones or vz езе not صا
wo y=
у х9 = әсе eui Jo javeuued ou; (г)
g--xc
v ge-g-X
3
sels)
901 (p)
هام
шо02 = 2х (7+9) =
өзед әш jo гајәшәд өц (z]
On جع عم و
ليهو مل
ЕСЕТ
vsx" fsx: sxe
seusxes 259-xc OD
ys)
[PI] 09€ (s)
И zi
эш) ü-:s360
o) Рм!)
si) oazis)
4% = .09¢ x = uum] pru
Jo efu دنر jo arrstou ey,
эйс B. uie pueros
Paue рдо jo aisea ou
reri
лиз jo insta au, (s) 9600 مر
beg < e
pai stieg әд jet Amngeooud ou] 20
а IMG өлеш Anqeco.d eu]
p = 0н 8 е9 әд мд Дуудецол! ey
Ê pasi yeq oaie Янаесо.л! ou t)
5
2
gg.
ЕС ce AN
249091 = à x Qr = ewe jesse) out
axem c ы шы
«un osez (и)
jew) ozta)
ots)
= ос бу
مر sifue janvoo Јо anew эці (sz)
д1 «у Gare эш (са) = مزح قناز
(о) ssou ©
مدع لج
P Ў
Eaters
эшацешшех+ [eut jo Siamsuy
هذا العمل خاص بموقع ذاكرولى التعليمى ولا
ty
وله على مواقع أخرى 1555551
рисоәз (8) (51336)
є-( 961 (2)
єс) «ое (н)
ole)
Bis)
zu
0с Sh = sou = مد
jo обие jeguan jo eunswow eu]
ia ae аа
10 биз enumo jo amewo ou.
بي هذا العمل خاص بموقع ذاكرولى التعليمى ولا 2
el n کس
suoneujwexg jeu! jo sjamsuy
TEE] على مواقع أخرى ayy
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